We have seen Atlanteans in both Marvel comics and DC. Beings who can live and breathe under water as well as on land, their physiology has been shown to be very different from humans.

Sticking my question to DC only, how do the Atlanteans actually communicate with each other under water?

They just talk.

With a mouth and everything, like you and I would.

In the comics, they just talk.

Leaving aside 60+ years of comics panels where Atlanteans speak with an open mouth, shout with a wide-open mouth etc, let us consider the "American tidal" arc (Aquaman #15-#20, 2004).

In this arc, the villain of the month is someone genuinely convinced that due to global warming, ice caps melting and the like, the whole world would eventually become sunken under water, and that the land would be gone (he's wrong, by the way). So, in order for humanity to be able to survive then, he engineered a genetic anomaly that he introduced in San Diego's water supply. The result being, dozens of inhabitants were unexpectedly "modified" to be able to live underwater (but they couldn't breathe air anymore).

Since this genetic anomaly was based on Aquaman's DNA, and that generally speaking Aquaman knows how to guide the newborn "water breathers", we can assume that their abilities are shared with those of Atlanteans.

Yeah, cool. When do we get to the Atlanteans talking part?

Well, in Aquaman #17, Lorena Marquez, one of these modified citizens, joins Aquaman underwater, is unsure how to speak... And Aquaman just tells her that she can talk down here. Atlanteans will just hear the sonic waves of her words thanks to special ears.

Then, it's not a matter of Atlanteans talking, it's about Atlanteans hearing.

I don't know. She's got bubbles of air coming out of her mouth, see? Okay, maybe Aquaman doesn't because he's a more genuine Atlantean than her, but maybe she could just talk because of the air in her lungs?

You're right to point that out! Right before this scene, she was still out of water, so her lungs would still have been filled with air. However, later on she breathes water, and can still talk, even if her lungs are filled with water. Starting from the next panels, she doesn't have air bubbles coming out of her mouth again.

We know they have water in their lungs because in Aquaman #22, Aquaman stops Mortimer Coolidge, a Sub Diego (the sunken part of San Diego) villain, by controlling the plankton contained in all the water Coolidge had inhaled.

Lol, defeated by plankton. Talk about a shame. Hey, yeah, that definitely indicates that their respiratory chords and stuff had evolved to handle water. My pulmonary system has worked great so far, but if my lungs are filled with water and I try to talk, that will be a mess, you know?

In Aquaman #15, Aquaman and scientists first discover that the modified people were water-breathers, by slicing open one's throat during an autopsy.

Aquaman describes this intricate gear-like throat organ as "not gills exactly, but something very close". Surely this enables them to proceed the water flowing out of their lungs into the "sonic waves" Aquaman was talking about above.

In the 2018 movie, they also just talk.

When Justice League (2017) came out, some people were upset with the only scene featuring two Atlanteans (Arthur and Mera) talking to each other, because it took place in an air bubble. There can be multiple explanations, ranging from "Mera wanted to speak privately" to "it was easier to shoot" - irrelevant.

Having now seen the movie (but unable to provide a clip yet), I can confirm they "just talk", although sound effects were obviously used to reflect the underwater setting.

Actually, when Aquaman takes his first Atlantean-style swimming lesson, Vulko explains it is a normal feature of Atlantean composition. Quote courtesy of IMDb.

Vulko: The ocean is not just our home, it is who we are.

Young Arthur: You can talk underwater!... Wait, I can talk underwater too! This is awesome!

Vulko: We can do so much more than talk. Being Atlantean is more than just being able to breathe under water, your body is equipped to survive in the extreme cold and pressure, and your eyes adjust to see in the darkness.

The following was written prior to the movie's release.

In the solo Aquaman movie (2018), director James Wan promised on Twitter that no such thing would happen.

Even more, Entertainment Weekly got to see a preview of the movie, and the only non-standard communication is some visual effects to suggest sound waves.

EW got to see about 15 minutes of the new film and noticed something. The characters … when they’re underwater … they just …

“People are overthinking it,” Wan says. “They’re just gonna talk!“

Precisely. Characters underwater speak normally in the film despite being underwater. Wan then adds a tad bit of a visual effect as Atlanteans speak, suggesting sound waves, but that’s all. It’s a rather brilliantly simple and naturalistic-looking solution to a problem that isn’t wasn’t actually that big of a problem in the first place.

_{"Aquaman director James Wan reveals how he’ll show underwater dialogue", Entertainment Weekly, June 13, 2018}

This might not always have been the case, though. Five years ago, when the Justice League movie was announced, Mark Millar gave an interview to SciFiNow, and he stated that Aquaman couldn't even talk in the water.

“You can get away with stuff in comics that in live action’s just a bit sucky – the best one is definitely Aquaman. Aquaman can’t even talk under water. If you think about it in comics it’s fine, you just have a speech balloon, but how do you have Atlantis and people talking under water? Are they gonna talking telepathically? Is it going to be body forms? The actual logistics of each member of the Justice League is disastrous, and you put them all together and I think you get an excellent way of losing $200 million.”

_{"Justice League film is “a good way to lose $200 million”", SciFiNow, February 5, 2013}

Now Mark Millar isn't exactly a noob when it comes to comics, so maybe in the comics he read in his youth (70s-80s) did clarify that Aquaman couldn't talk. So far, I found nothing of the sort.