I don't see how this is any different to having access to the forums. Why would it be a "moderation and privacy nightmare"? (See my notes below.)
Unless the current moderation for the forums is already too much of a challenge. Private messages are posts, too, as far as moderation is concerned. They can be shared with moderators when reported. And the same policies and user treatment apply for the most part. An offending user can be warned, then banned.
It's no different to using the forums. The only difference is that users can talk to each other without everyone having to ogle their posts.
Viewing post in Feature Request: Private Messages
Since you necroed this topic anyway, have you seen how people argue about DMs on Mastodon? Or for that matter the problems that crop up all the time on Discord? No, DMs are not the same as an open forum. There are different expectations, and different issues, precisely because DMs aren't public. Sure, you can do all that. But it can be a lot more problematic.