Is private the right word? I don't think they were trying to share their conversation with the rest of the room, but they also didn't go very far in trying to keep it from the rest of the room.

In a legal sense, private more or less goes out the window when you share something with a third party, a court would be unlikely to step in and protect the privacy of a conversation held in a shared space.

The internet conversation certainly doesn't have to use the legal definition, but "private" does sort of invoke the idea that the communication is intentionally being held close, and if you are talking in a room with other people, you aren't working very hard at that.