I have not voted, since I am too much in doubt what would be best.
Messages between Admin/Mod and Users I would approve of; although that can also be handled by email, a PM might be quicker/more direct.
Some topics have a lot of remarks that are mostly meant for a single person, and especially if they are off-topic, I would like that those would be done via PM, like @Max9 suggests. Occasionally I have wished I could send someone a PM, because the message or reply could be sensitive, or not suitable for all members (esp. minors).
But I also fear abuse, and am not sure how easy it is to stop that / keep that under control. Obviously PM should only be available to members, but I suspect you couldn't even implement it for non-members, so that is simple.
Granting it to only some members quickly will lead to protests from those who do not get it, unless you have strict and clear guidelines who gets it. I can imagine no minors (for their own protection), and perhaps a probationary period like @JonesMacGyver suggested, although perhaps 1 month should be long enough. A minimum number of (public) posts could also be an option, but that might lead to a lot of irrelevant posts to quickly reach that threshold.
What you sometimes also see is that PM is only available between 'friends' - that way the receiver can decide beforehand if PM's would be welcome from a certain member. But that would also have to be available in the forum software and needs to be implemented and maintained as well. If this were possible, I'd vote Yes.