Actually, the private message function has not worked in the past. I can't even remember the last time I bothered trying to use it.
Having a Yedda contact can be a way to follow that person's questions and/or answers.
For instance, if you came across someone whose answers you felt were consistently wrong (in fact, harmful) then you might make that person a Contact as a means of being able to see which questions s/he had answered so you could post a corrective reply.
In other words, rather like doing damage control.
Contacts also come in handy if you come across questions you know that person can answer.
For instance, let's say you typically answer Yedda questions regarding relationships. In doing so, you keep running across someone whose advice on relationships is consistently good.
If you make that person a Contact, then when you run across a relationship question you think that person can answer, you merely send the question to him or her using the 'Send to a friend' link you see at the bottom of each question.
That's mostly what I use it for.
There may be other uses, but I haven't discovered them yet. I AM still relatively new - I've only been a Yedda member for four months!