On one hand, having a group chat of some sort to discuss hero-ing shenanigans seems useful. On the other, using Discord or even a text group chat seems like a pretty good way for some tech company to get their hands on a whole bunch of secret identities. Do superheroes have a proprietary app for this sort of thing? Or do they have another solution?
For the JLA and Titans-related heroes, Cyborg's hacking powers render this a total non-issue. On the client end, they use the same texting and messaging apps as everyone else, with a few added bits and bobbles of interface.
On the server end, software that Cyborg set up (took him about 5 minutes) casually intercepts every message they send before it reaches any servers corporations could extract data from, redirects to a private, secure instance of the same chat/texting software (getting Facebook's total source code without authorization is about 0.02 seconds of effort for him), also redirects anyone else in the same chatspace to the same private server, then filters all the messages for sensitive info (there's ways people can flag their own messages) before hacking back into the corporate servers to re-insert a record of some messages they would expect to see as if their own servers had handled everything, so even if admins try to investigate they won't be able to tell who's using the pirated chat instance to ban them.
It's all very complicated from a human perspective but it's all happening totally automatically in real time so it doesn't really matter. The end result is the Titans just have a regular old Discord server that, as far as Discord the company is concerned, doesn't exist, but they can still invite people to who are using regular Discord accounts or chat from their accounts on other normal servers, and other apps are all similarly seamless and convenient.
The Avengers have basically an ultra-high-end spy version of Signal (the app, not Batman's latest son). At first they were just using SHIELD messaging software but after not too long Stark made his own. But the way it's set up, not even Stark has access to all the messages that are sent to anyone besides him. During Civil War he tried to shut down the whole service so the other side couldn't use it, and failed. He just made it too secure and autonymous. Unless someone has the app open and is currently looking at what he wants to see, not even Cyborg could extract chatlogs from it.
The X-Men, X-Force, and X-Factor are basically left out in the cold and forced to use a combination of regular services (some encrypted like Signal) and, when available, and only for the people who agree to it, telepathic chatrooms.
Whoever is "hosting" a telepathic chatroom isn't consciously aware of the messages being subconsciously relayed in real time, but, there's nothing to stop them browsing if they get curious, and as soon as they need to focus and use their powers on anything else strenuous or with stakes, the whole thing shuts down. But the chatlogs being stored in the literal memories of the people in them, that only prevents new messages from being sent for a while (and from being able to literally visually see it), and if the telepathic chatrooms need a new "host" to run it, little is lost in the hand-off.
Of course all of that is just about text chat. Earpiece comms are fairly standard among superheroes, so a lot of communication happens over comms voice calls/recorded voice messages, and that's its own whole topic.