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Motivating scenario:

I'm a swell upstanding guy, so Ser/Lord X, an acquaintance of mine, knights me. So I walk into my village tavern; the barkeep asks "What'll it be, friend?" and I tell him "That's 'Ser' to you now. I have been knighted by X!" He sneers and goes "well then, X can call you Ser if he likes."

Actual Question:

What about a person's being knighted makes the knighting 'stick'? Is it the identity/standing of the knighter (is that a word?) ? Is it the fact that many other knights walk around telling people that the new guy is also a Knight? Is it the public nature of the knighting?

The Wiki of Ice and Fire says the following about knighting:

This usually happens when a squire reaches adulthood and his master
judges him worthy of accepting the responsibilities of a knight. A man
who has not been raised in the knightly tradition can also be made a
knight as a reward for service. This is often granted to soldiers or
other smallfolk who have shown bravery or performed a great feat.

Notes:

• I'm only asking about Westerosi Andal-culture Knights, around the time of Robert I Baratheon's rule.

tl;dr:

In theory, a Westerosi knight gains is title when the knight he is squired to deems him ready, or when he does something to gain the favor of a lord or monarch (almost always militarily.)

In practice, anyone can basically call themselves a knight, and as long as they look and act the part (which basically boils down to having a horse and armor), and people are willing to call them Ser, that's all that matters. The more well-known a knight becomes (serving powerful lords, having impressive military service, etc) the more likely people will be to accept the title at face value.

Historically speaking, the term "knight" has evolved over the course of the Middle Ages to mean several very different things. Early on, a "knight" was essentially any well-equipped, mounted soldier fighting for his monarch, lord, the Church, etc. Over time, knighthood turned into a sort of minor nobility, and the title become more and more official. It was during this time that the concept of "orders" or knights that followed specific codes of conduct, and more generally the code of chivalry, gained favor. By the time of the War of the Roses (the period of English history that Westeros is roughly based on), knighthood was a very formal title, usually granted by the ruling monarch, Pope, or other very powerful political figure, and came with official documents (letters patent) granting the title. This is basically where things stand today, apart from the fact that the title is purely honorary, independent of any military service.

Westerosi culture seems to be stuck very early in this progression of knighthood, where there were no official orders of knighthood and no central authorities bestowing the titles on people. The title of Ser in Westeros seems to follow the same basic process as the craft trades: a potential knight is handed over to a current knight as a youngster to become a squire (an "apprentice knight"), learning the craft until he gets good enough to be declared a knight of his own. In this sense, anyone who is currently recognized as a knight can declare their squires knights, in their own right, when they feel they are ready.

Of course, as this is basically the honor system, it's rife for abuse. No central authority ensures that the so-called knights meet any particular qualification, and there's no real set of rules they have to follow. Oddly, it seems that the popular ideal of chivalrous knights has arisen in Westeros, but without the practical reality of chivalrous orders to back it up. And, other than the ruling monarch stripping someone of their title (which I don't know that we've ever seen), once a knight, always a knight.

On the other hand, since being a knight has no "official" meaning in Westeros, people are conditioned to just assume any well-equipped, mounted soldier, especially one that serves under a king or lord, is probably a knight. And as long as people are willing to grant that title to those soldiers, and they are willing to accept it, that's about all that matters. This is why The Hound is so adamant that he's not a knight: he both looks and acts the part, so anyone meeting him will almost immediately assume he is one, until he claims otherwise.

From the words of the Author

George R.R. Martin says in an e-mail correspondence that it's "social pressure" which keeps someone from lying about being a knight:

[Interviewer] Yet if every knight can create a knight as we have seen in "The Hedge Knight", what prevents a widespread misuse by unscrupulous hedge knights such as Ser Osmynd Kettleblack or the cynical Great Houses?
[GRRM] Social pressure. A knight's peers would look with a certain amount of disfavor on anyone who did this. They might gain money, but they would lose honor. And honor is still very important in this culture.

There's more in the interview on the subject, but this is the core of it.

Repute

We know that any knight can make a knight¹, but to make it 'stick' the knighting knight needs to have the appropriate renown, as explained by the greatest knight of all, Ser Barristan "The Bold" Selmy:

^{[A Dance With Dragons:]}

As he watched them at their drills, Ser Barristan pondered raising Tumco and Larraq to knighthood then and there, and mayhaps the Red Lamb too. It required a knight to make a knight, and if something should go awry tonight, dawn might find him dead or in a dungeon. Who would dub his squires then? On the other hand, a young knight’s repute derived at least in part from the honor of the man who conferred knighthood on him. It would do his lads no good at all if it was known that they were given their spurs by a traitor, and might well land them in the dungeon next to him. They deserve better, Ser Barristan decided. Better a long life as a squire than a short one as a soiled knight.
^{-A Song of Ice and Fire: A Dance With Dragons, Part Two - After the Feast (The Kingbreaker).}

Examples

1. 
   Ser Duncan The Tall is featured in the novellas The Dunk and Egg Stories. Dunk himself has difficulty proving his knighthood, but in the end he got vouched for by Prince Baelor "Breakspear" Targaryen.


2. Also from The Dunk and Egg Stories (The Mystery Knight), Ser Glendon Ball claims that he was knighted in front of many witnesses².

Behaviour

A Knight is generally one who is chivalrous and upstanding³ - barring the few who use the title as a means to a gains - this is what sets them apart from normal men (as well as the armor and weaponry.

We ride spinners

A small point which I picked up is that people can almost tell someone is a knight from the horse they ride:

“Salt pork never pleases me.” The pitchfork knight gave Clegane only the most cursory glance, and paid no attention at all to Arya, but he looked long and hard at Stranger. The stallion was no plow horse, that was plain at a glance. One of the squires almost wound up in the mud when the big black courser bit at his own mount. “How did you come by this beast?” the pitchfork knight demanded.
...

“Because knights are fools, and it would have been beneath him to look twice at some poxy peasant.” He gave the horses a lick with the whip. “Keep your eyes down and your tone respectful and say ser a lot, and most knights will never see you. They pay more mind to horses than to smallfolk. He might have known Stranger if he’d ever seen me ride him.”

^{-A Song of Ice and Fire: A Storm of Swords (Arya).}

The talk of the town

The fastest way for any news to spread is via gossip. I assume it's not too far-fetched to imagine that the news of a knighting spreads quickly around the realm; imagine something like a bunch of butchers hanging out and chatting among each other "Did you hear that sellsword Bronn got knighted? I saw that the High Septon himself dabbed him and he was dubbed by the Kingsguard!" ... "Wow I need to tell Johnson from Hightower about this!"...

LMGTFY

Unfortunately, there is no possibility of Googling in Westeros, or Essos for that matter, so people can't just go "oh you say you're a knight? let me just check that on the Internet Knighthood DataBase (iKDB) ... Yup, you check out".

There is however the fact that each Knight and House has a coat of arms, and are therefore distinguishable. These are often recorded in books known as A "Roll of Arms".

^{2. From Dunk and Egg Stories, The Mystery Knight:}

Half a year ago, however, a party of knights chanced upon the brothel and a certain Ser Morgan Dunstable took a drunken fancy to Ser Glendon's sister. As it happens, the sister was still a virgin and Dunstable did not have the price of her maidenhead. So a bargain was struck. Ser Morgan clubbed her brother a knight, right there in the Pussywillows in front of twenty witnesses, and afterwards little sister took him upstairs and let him pluck her flower. And there you are."


Any knight could make a knight. When he was squiring for Ser Arlan, Dunk had heard tales of other men who'd bought their knighthood with a kindness or a threat or a bag of silver coins, but never with a sister's maiden-head."

^{3. From the wiki:}

Knights are supposed to be brave, courageous, honorable, and true to their word. They should be loyal to their feudal overlord, and are tasked with defending their faith. In practice, most knights fall short of sustaining such high and noble ideals. Many do not try particularly hard, and instead seek knighthood mainly for the prestige and opportunities.

^{4. See this conversation between Dunk and Egg:}

"Do you want a clout in the ear? What book is that?" Dunk saw bright colors on the page, little painted shields hiding in amongst the letters.

"A roll of arms, ser."

"Looking for the Fiddler? You won't find him. They don't put hedge knights in those rolls, just lords and champions."
-Dunk and Egg Stories, The Mystery Knight

Knighthood and Andal Culture

You don't need to specifically narrow down your question to Andals. Knighthood is an Andal institution. The Would-be Knights generally have to swear vows in name of the seven gods of Andals and they have to stand a vigil in a sept. Which is why Knights are very rare (But not unheard of) in Iron Islands & North. As evident from Knighting ceremony procedure:

a touch on the right shoulder with the blade. "In the name of the
Warrior I charge you to be brave."

The sword moves from right shoulder to left. "In the name of the
Father I charge you to be just."

Right shoulder. "In the name of the Mother I charge you to defend
the young and innocent."

The left. "In the name of the Maid I charge you to protect all
women....

Ironborn and Northmen do not consider these deities true therefore they do not usually get involved with Knighthood.

However, I just observed that a generic oath can also be used without specifically mentioning the deities.

A wan smile crossed Lord Beric’s lips. “Thoros, my sword.”

This
time the lightning lord did not set the blade afire, but merely laid
it light on Gendry’s shoulder. “Gendry, do you swear before the eyes
of gods and men to defend those who cannot defend themselves, to
protect all women and children, to obey your captains, your liege
lord, and your king, to fight bravely when needed and do such other
tasks as are laid upon you, however hard or humble or dangerous they
may be?”

“I do, m’lord.”

The marcher lord moved the sword from
the right shoulder to the left, and said, “Arise Ser Gendry, knight
of the hollow hill, and be welcome to our brotherhood.”
_{A Storm of Swords - Arya VIII}

What makes a Knight?

Any Knight can make a Knight. All you need is someone who is a Knight, he can then hear your vows and dub you a Knight. Ser Lyonel Baratheon said this to Ser Duncan the Tall in The Hedge Knight:

"The lad has the truth of it," said Ser Lyonel Baratheon. "Do it, Ser
Duncan. Any knight can make a knight".

That being said, your question is valid. Do people believe that a person was knighted? This is why it is customary to perform Knighting ceremony in front of Witnesses.

When Ser Duncan the Tall claimed to be a knight, The Steward of Ashford, a man named Plummer, instantly picked on that and pointed out he should have had his knighthood in front of a Septon and Witnesses. Quoting from The Hedge Knight:

“Hmpf.” The man Plummer rubbed his nose. “Any knight can make a
knight, it is true, though it is more customary to stand a vigil and
be anointed by a septon before taking your vows. Were there any
witnesses to your dubbing?”

There are four paths to obtaining Knighthood:

1. The Regular Path

This path is followed by the nobility and chivalry of Southern Kingdoms who are of general Andal descent and follow the faith of Seven. It involves following steps:

1. Become a Page of a Knight as a child. Run simple errands and be trained in basic martial Arts.


2. Become squire of a knight. Learn about weapons and equipment. Learn more advanced martial arts. Accompany your master to battle and fight if need be.


3. Be knighted by your master when he deems you are ready. You will have to stand a vigil in a sept and be anointed by a Septon but it is just customary, not required. Witnesses should be present but that is also only customary to maintain some proof of some person's knighthood.

2. Valor-in-Battle Path

The second path does not involve the long training of regular path. If you show extra ordinary valor in a battle, you can be knighted by your King or your Liege lord (Ser Jorah Mormont was knighted for his valor in Siege of Pyke during Balon Greyjoy's 1st Rebellion). If they are not knights, they might ask some other knight to knight you (Daenerys asked Ser Jorah to knight Ser Barristan Selmy, unaware of the fact that Selmy was already a legendary knight).

3. Buy-the-knighthood path

Third path is the easiest. You just give a knight something he wants and in return you can demand knighthood. Ser Glendon Flowers bought his knighthood from Ser Morgan Dunstable at price of his sister's maidenhead. Ser Morgan accepted the offer and knighted Glendon in front of a dozen witnesses.

4. Perform a great deed

If you perform a great deed for some noble, he may knight you in reward. Ser Davos Seaworth was knighted by Stannis Baratheon for saving the garrison of Stormsend from death by starvation.

In all these methods, One thing is common. No matter what path you choose, some witnesses are always customary. It is those witnesses who are then contacted if a real legal proof of someone's knighthood is needed. e.g. if a Knight is accused of a crime and demands a trial by combat, The accusing authority will be within its rights to demand proof of knighthood. The burden of acquiring the proof which is mostly someone vouching for their status is on the accused.

How do Common people react to such claims?

As already stated by KutuluMike, as far as common people are concerned, if someone is wearing an armour and riding a horse, they are okay with giving them the courtesy of calling "Ser". Brienne of Tarth was mistakenly called Ser many times despite not being a Knight. If you look like a knight and can act like a knight, you can claim you are a knight but it will be very difficult for you to prove if some proper authority demands a proof or testimony.

But if you look haggard and poor, even a horse and armor won't give credence to your claim. Knights are usually sons of Noble lords and thus are well-to-do. Hedge Knights do exist as well but they are generally viewed with suspicion. Quoting from The Mystery Knight, this is what an innkeep had to say about poor knights:

"The lake is full of fish and you will find some other rogues camped
down by the stumps. Hedge Knights, If you believe them". Her tone
made it clear that she did not.

So if you are unknown, it really depends on the other person whether they consider you a knight or not.

How to make it stick

There is only one way to do it. Be Famous. There are different ways of becoming famous:

1. Westerosi people love to follow tourneys and knights. Knights who show talent in tourneys are sort of comparable to celebrities in our world. So get your jousting game up. Barristan Selmy was only a child when he sneaked in to ride in a tourney. Prince of Dragonflies, Duncan Targaryen unseated him and gave him the epithet "The Bold". Selmy was known as The Bold for the rest of his life, even after his real knighthood.


2. Robber Knights are also sort of celebrities. You can become an outlaw knight and win commons like Brotherhood of the Kingswood did. You will be famous in no time.


3. Perform valiant acts in battle and form comradeship. If you fight well and maintain good relations with warriors coming from all parts of the Kingdom, you will likely create a reputation for yourself. When Jaime was investigating knighthood of Ser Kettleblack, first questions he asked were about the tourney Kettleblack had rode in and battles he had fought in.


4. Excel at skill at arms and catch attention of some lord using option 1 and 3. Eventually, if you are good enough, word will reach to the King and you might get a white cloak. Then your name will be known to everyone in the Kingdoms.


5. Make sure that you do get a decent number of witnesses to view your knighting ceremony. If you do, at least in that locale you won't have any troubles with getting the title "Ser" from peasants.

If you are famous, no one will doubt your knighthood.

For example:

Barristan Selmy goes to an inn.

Barristan: Hullo, I am Barristan Selmy.

Innkeep: Seven hells, It's Ser Barristan the Bold. I gotta get a
portrait drawn with you, Ser. Drinks on the house.