I really think it is a good idea to be able to send private messages. There are many things someone might want to talk about, without making it all public. We also don't want to give out our email in public, but might do it with a private message, when we get to know someone better. It's a good idea.
Because it wouldn't be used as such. People would just send messages back and forth like its for talking to your friend. One thing I'd like ask to do though is alert you by email when someone comments on your answer.