I'm an old Usenet hand--about 10 years, so it's since the Great Renaming, and admittedly a lot of that has been lurking, but point is I've been around online forums (fora?) for quite a long time and like to think I more or less know where my towel is. What one can and can't post of private email in a newsgroup is more or less analagous to PMs and Mudcat threads, so here goes:
Of course, the first thing to realize is that no (or at least almost no) digital communication can be guaranteed to remain private. It's not illegal for someone to post in public an email sent to them in private, but it is considered very bad netiquette. So don't assume that any message you have sent can be kept private, and tread lightly--very, very lightly--around even considering posting the private words of another. There's no official, legal censure to it, but there is a strong personal and social one.
As for messages you yourself have sent in private, I'd say you're more free to post their contents, since you own the words. People have done this in e-forums for eons. Of course--and especially if the issue at hand is fairly heated--this can put the other correspondent into the position of having to post his or her own private words if he or she feels it's the only way to refute/explain/counterattack/whatever. So, posting your own private words, while more socially acceptable, can easily become a way to out someone's private words to you without posting them yourselves, and for that reason should be treaded around lightly as well.
I am aware of the specific issue that is at concern for you here (as aware as can be from what little's been said about it in other threads I've skimmed), but I hope that that what I'm saying here isn't tainted either positively or negatively by those very real issues. I seek to answer your query at as objective, neutral, and theoretical a level as possible.
I hope it helps.