But chat is private, like your messages.
I don't know how the discourse chat works, but that's not what I understand by chat...
Only private chats are private, there are also public "rooms" - the difference with chat is just that it is more immediate, lots of shorter items, more conversational, and not threaded (by topic). Just split into category based public rooms or private rooms (which can be one to one).
So maybe discourse chat is a bit different.