I know this is a common feature on social networks and I am the last person who wants EL&U to resemble Facebook, perish the thought, but I believe a system where one user can communicate privately with another would be a useful tool.
Chat is not the same, and the interface is positively a mess. Personally, I can't make head nor tail of it and find, despite its name, to be contradictorily intimidating. Chat is public, anyone can join a conversation (invited or not), but if a trusted user has a query, would like some advice on formulating an answer or has a question that is "too localized," then why make it public?
"Why not? It might be of interest to others" someone might chime in. Not if it is too localized and furthermore, not every legitimate question on grammar, syntax, lexis etc... is intrinsically interesting. For instance, a question I posted recently was really bugging me, I spent ages trying to make it look more "challenging" and worthy of a response. Thankfully, I did get one, and I was very grateful but it was clear as questions go, it was a resounding flop. I'd delete the question, but I can't because there's an answer! How much easier it would have been with a PM.
Private messages would also help resolve discussions and disagreements. They might also cause a few new ones too! However, the majority of users on this website are mature adults, and I think we can be trusted.
There is also a final advantage I would like to mention. It would help avoid unfortunate incidents where an OP or a respondent misinterprets the intervention of some of EL&U's most respected experts. Such as illustrated by this comment by John Lawler. In my opinion, people's egos are fragile things and more often than not we are aware of our own shortcomings. It takes courage to admit publicly that a long-held belief (and many grammar rules learnt at school are the cause of such incomprehensions) is at best inaccurate or worst a complete and utter lie. Not everyone can admit to being wrong, and certainly it is unrealistic to expect people to change their opinions in a heart beat, even when faced with indisputable evidence. A PM sent by member gently pointing out an error or asking for clarification might be listened to more carefully.
EDIT: I've just realized that my last paragraph seems to uphold J.Lawyer's conviction. Not so. PMs would avoid users' interpretations of being singled out for criticism on this platform. I like having a wide choice of answers to choose from, sometimes the low-voted answers help me as much as the most popular ones.