I had a conversation yesterday with a user who couldn't see all the content due to workplace firewall rules. (no imgur allowed - but pastebin is, go figure)
He sent me his email in a me-him chat room, which apparently still isn't private.
Fortunately a Moderator notice what was going on and deleted the personal information after asking.
This could be solved by implementing a whisper function. Whisper allows private messages between two users currently in the same chat room. This is typically invoked in IRC like:
/whisper @UserID Message_goes_here
And @UserID would see my message in the normal chat window with distinctive formatting.
Chat is preferred to an email proxy or message box as it is opt in - I have to be in the chat room, and can leave at any time, closing the connection.
Whisper is also preferred as it is clunky, less likely to be used for ongoing conversations. It is to some degree self limiting.