I feel there should be a message system where you can leave specific users private messages, so that users can comunicate privately, which is why the chat won't do. How come we don't have anything like that yet? I know S.E. doesn't have it as a feature in any site, but, hey, it's never too late.
I can't find a specific link or quote right now, but afaik that was specifically left out because SE doesn't intend to be a forum/chat community but rather a dedicated Q&A site. IIRC the tenor was that including too many social features (like private messages) would diminish or dilute the main focus of the sites.
Communication between users should be focused on the questions and answers and comments and the chat already provide that functionality. If users wants to be able to discuss other things they can always include further contact information (email, facebook, whatever) in their account profile.
Most basic questions have been addressed network-wide on Meta Stack Overflow. Answer to this one is "no." Any way to send a personal message to another user?
The consistent response to this is that it's a bad idea, because:
It could hide information from the community: useful information transmitted privately is unavailable to other readers, subverting the core purpose of the site.
It could be used to harass other users ("Answer my question!", "Accept my answer!", "Yer momma so fat she overflows the stack!", etc...)