No attribute-to-damage with the second weapon, unless you have the appropriate fighting style.

I suspect you're seeing the first bullet point of Martial Arts...

You gain the following benefits while you are unarmed or wielding only monk weapons and you aren’t wearing armor or wielding a shield:

• You can use Dexterity instead of Strength for the attack and damage rolls of your unarmed strikes and monk weapons.

...and thinking it overrides this clause in Two Weapon Fighting:

[...] You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative.

Martial Arts allows you to use Dexterity for things that are normally Strength based. However, it doesn't have any text that overrides the two-weapon fighting rule saying you only apply negative ability modifiers to the second attack's damage. There are no secret rules, no hidden combinations. If Martial Arts was meant to override the lack of an ability modifier on damage specifically cited in two-weapon fighting, it would say so... like the Two Weapon Fighting style does. The

When you engage in two-weapon fighting, you can add your ability modifier to the damage of the second attack.

To put it another way/in summary... If you want to get the benefit of something (Two-Weapon Fighting Style), you have to actually take that thing.

Two-Weapon Fighting Style and elements of Martial Arts can be used together, but not by a Monk/Rogue.

If the character was a Monk/Fighter¹ with Two Weapon Fighting style and the Dual Wielder Feat (because a quarterstaff is not light), using a quarterstaff as the second weapon they could use Dexterity on the second attack (via Martial Arts) and add Dexterity to damage (via Two Weapon Fighting Style).

...of course, I'm not sure why you'd want to make such an investment for very little gain, but you could.

Extra Attempts at Sneak Attack

I suspect you're trying to get more attack rolls that can potentially trigger a Sneak Attack. Good on you if you're trying to using a finesse weapon because you realize Martial Arts is not the same thing as a weapon having finesse, even if the effect is very much similar. Just remember that Sneak Attack is only once per turn. They won't all be Sneak Attacks.

Note that you don't have to decide which bonus action to take ahead of time.

• If the initial shortsword attack misses, you can use the second shortsword to take a swing as a bonus action via two-weapon fighting. It won't get attribute-to-damage as explained above, but it does give you a second chance to trigger Sneak Attack. With enough rogue levels², it's worth the sacrifice.


• If the initial shortsword attack hits and triggers Sneak Attack, you can use Martial Arts to make an unarmed attack with your bonus action, which would get attribute-to-damage. By having a shortsword in both hands, your initial attack does not violate the conditions of Martial Arts.

¹_{...or Ranger or College of Swords Bard}²One rogue level gets you an average of 3.5 damage, which would be better than the 3 damage from DEX 16.

No

How you use your Bonus Action is subject to the specific rules associated with what you're trying to do.

If you are using the rules for Two-Weapon Fighting, then you may use your Action to make an attack with a short sword. This attack will deal 1d6 damage plus your Dex modifier because it is a finesse weapon. You may then followup with a Bonus Action attack that deals 1d6 damage, but does NOT include your Dex modifier.

If you are using the rules for Martial Arts, then you may use your Action to make an attack with a short sword, which is a monk weapon. This attack will deal 1d6 damage plus your Dex modifer because is it a monk weapon. You may then followup with a Bonus Action unarmed strike attack that deals damage equal to your Martial Arts die in the Monk table; this DOES include your Dex modifier.

This is a scenario where there are 2 sets of rules whereby the character may utilize their Bonus Action. However, you cannot mix the criteria by picking the most beneficial from two-weapon fighting and the most beneficial from Martial Arts.

You cannot add your Dex Mod to the 'Two Weapon Fighting' damage roll, using only Martial Arts

The wording of the 'specific' rule does not override the general rule that Two-Weapon-Fighting (the action, not the fighting style) attacks do not add your Dex Modifier to the damage roll.

The wording of that bullet point is this.

You can use Dexterity instead of Strength for the attack and damage rolls of your unarmed strikes and monk weapons.

Source

It does NOT say "You can add your Dexterity to the damage rolls of your unarmed strikes and monk weapons."

What it is saying is "If a monk weapon normally forces you to use Strength for your attack and damage rolls (like a Quarterstaff), you can choose to use Dexterity instead."

Compare that to the wording of the Two Weapon Fighting Action

You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative.

Source

And the wording of a Fighter's "Two Weapon Fighting" fighting style

When you engage in two-weapon fighting, you can add your ability modifier to the damage of the second attack.

Source

Also, for comparison...the Finesse property of weapons...

When making an attack with a finesse weapon, you use your choice of your Strength or Dexterity modifier for the attack and damage rolls. You must use the same modifier for both rolls.

Source

Notice the similarity between Finesse and Martial Arts. The Martial Arts Feature has the same basic effect. With these weapons, you may use Dex instead of Str for your attack/damage rolls. It does not let you override the "No Ability Modifier added to damage" rule of the Two Weapon Fighting action.

Your second attack uses general two weapon fighting rules(Ch-9).

The Martial Arts feature is explicit about the special nature of the second attack in bullet 3.

When you use the Attack action with an unarmed strike or a monk weapon on your turn, you can make one unarmed strike as a bonus action. For example, if you take the Attack action and attack with a quarterstaff, you can also make an unarmed strike as a bonus action, assuming you haven’t already taken a bonus action this turn.

It does not say "you can make one unarmed strike or a monk weapon attack" as a bonus action. Since you are not using that special Monk feature, in that your question says "instead of bullet three" you revert to the general case of two-weapon fighting. (For "specific beats general" see below)

Any PC can use generic Two Weapon Fighting from Chapter 9 which specifically removes the dexterity bonus from the second attack.

Two-Weapon Fighting

When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can use a bonus action to attack with a different light melee weapon that you’re holding in the other hand. You don’t add your ability modifier to the damage of the bonus attack, unless that modifier is negative.

If either weapon has the thrown property, you can throw the weapon, instead of making a melee attack with it.

Naming convention: WoTC Arrgghhh! moment

This is a common point of confusion due to a name choice. The general two weapon fighting versus two weapon fighting style. Only the Ranger and the Fighter get the Two Weapon Fighting Style.

Two-Weapon Fighting

When you engage in two-weapon fighting, you can add your ability modifier to the damage of the second attack (p. 27, Basic Rules)

It is unfortunate that they used the same term for two subtly different features. One is unique to two martial classes, the other is usable by any PC/NPC. If they could have called the fighting style "Two-Handed Fighting Style" to differentiate it from the general "two weapon fighting" this confusion might be alleviated.

Specific beats General: does something break if Monk gets this benefit?

Hard to say. We are talking one average damage difference for a pure Monk at low levels, and a hope to trigger a Sneak Attack (and keep bonus damage) for your multi-classed Monk/Rogue(Arcane Trickster). Since you did not clearly identify how many levels of Monk and how many levels of Rogue(AT) you are using, this assessment can't cover all cases.

This book contains rules, especially in parts 2 and 3, that govern how the game plays. That said, many racial traits, class features, spells, magic items, monster abilities, and other game elements break the general rules in some way, creating an exception to how the rest of the game works. Remember this: If a specific rule contradicts a general rule, the specific rule wins (Basic Rules, p. 5)

Class features are in Part 1 of the rules, and those specific rules beat general rules.

When appealing to "specific over general" which rule is specific and which rule is general? (T.J.L has offered one analysis on this in his answer, in terms of the Monk rule not being specific enough as compared to the Ranger/Fighter fighting style). So too does guildsbounty, in greater detail

• 
  

  Are these two specific rules overwriting a general rule about weapon to hit and damage bonuses, and or the general two weapon fighting rules?

  
  
  
  
  
  • Is the Class Feature (Monk) more specific than the General Feature (Ch 9, Two weapon fighting), or is it the other way around? The Fighter and Ranger Class features certainly are more specific than the Chapter 9 rule regarding two weapon fighting.
  
  
  
  

... many racial traits, class features, spells, magic items,
monster abilities, and other game elements break the general rules in
some way

You can make an argument that the specific class feature of Monk breaks the general Two Weapon Fighting Rule that applies to all other classes, or, you can argue that the Monk rule is a variation on the finesse weapon rule that applies to weapons other than light / finesse, such as clubs, hand axes, quarterstaffs and maces. (T.J.L. seems to make the latter point). All simple weapons (that don’t have the two-handed or heavy property) are Monk weapons.

You can use Dexterity instead of Strength for the attack and damage rolls of your unarmed strikes and monk weapons.

The argument is that what makes this Monk class specific rule overwrite a general rule is that all simple weapons are Monk weapons(that's not a short list) so now a Club, Handaxe, Javelin, Light hammer, Mace, Quarterstaff and Spear, all use the Dexterity bonus for attack and damage. This provides a Monk a flexibility not available to any other class in terms of applying Dex where Str is usually the case. (And if an improvised weapon is ruled as being a club, then at higher levels that club becomes a monk weapon and its weapon damage increases as monk damage increases ...)

Compare Finesse to the Monk class feature.

Finesse. When making an attack with a finesse weapon, you use your choice of your Strength or Dexterity modifier for the attack and damage rolls. You must use the same modifier for both rolls.

The ruling needs to come from the DM since it is not clear which is the specific rule that supersedes the general rule. If you really want this, then I recommend that you make this appeal to your DM on the basis that the Monk class feature is more specific (as a class feature) than the general rule on two weapon fighting (Ch-9) since short sword is a Monk Weapon by that class feature. Be prepared for a ruling that looks at it the other way: that the Monk class feature isn't specific enough and is specific over general in a different sense: some bludgeoning/slashing weapons get Dexterity bonuses to hit and damage where normally Strength is the ability bonus for that.

At first level applying this ruling would put the monk in the same damage output range of a two weapon fighting style fighter. At second level, this matches the Ranger with that choice. By the time your Arcane Trickster arrives at level 4, (assuming 1/3) the other two have an ASI and you don't have one yet.

At level 5, you either get an ASI or a feat or another level of monk (ki) and the other two classes now have two attacks.

A note on the two other martial characters, Ranger and Fighter, who have to choose their two weapon fighting style at the expense of Archery, defense, Great Weapons Fighting, etc. A problem can be that, with this ruling, the Ranger or Fighter had to make a sacrifice (pick that fighting style over any other) while the Monk had to make no such choice. That may be perceived as a problem at your table by other players, and it may not. Discuss this at your table and with your DM.

By 5th level your Monk damage for the unarmed attack matches the short sword and renders it moot, and at 6th level all of your unarmed strikes are magical attacks. That may be more important, getting that hit against resistant and immune monsters. Discuss with your DM; I'd be inclined to rule in favor of it unless someone at the table perceived that your character was getting an advantage they did not.

Multiclassing

Sneak Attack damage, as a Rogue, tends to overwrite the Dexterity bonus as you gain levels in Arcane Trickster. By the time you choose AT, sneak attack is 2d6. At level 5 it is 3d6 and so on. The higher the level the less important that dex bonus is to your damage bursts.

You only trigger Sneak Attack once on your turn, so you can't get that 2d6 twice if you hit twice; you can get it once if you hit once in two attacks. Even at low level, one trigger of sneak attack damage is typically more than the Dexterity bonus.

How long will this game last? As your Rogue levels go up, and sneak attack damage goes up, so the burst damage opportunities go up.

No this character does not add their Dex bonus to damage to an offhand attack in this case.

The Martial Arts rule is very similar to the rule about finesse, i.e. it allows a character to use it's dex modifier rather than it's strength modifier in weapon attacks, except it applies to a different set of weapons defined as "unarmed or monk weapons" rather than "weapons with the finesse feature".

In the same way that finesse does not allow you to apply damage to a bonus action offhand strike, nor does the Martial Arts feature.