I understand that there is Flagrow byōbu and I admire the work that luceos is done. And I really see the beauty in reusing and extending main discussions for private conversations.
But this is an example of how a programmer would approach the problem. I think that a typical forum owner would never do (or want it to be done) this way. As a forum owner, I want to encourage public conversations. "Private messages" is a necessary secondary feature, but it should not clutter the mainstream activity and distract users from creation of public posts. Byōbu extension turns Flarum into a messaging platform and it just does not feel right.
So, since in other threads people are looking for ideas of an extension to make, I would suggest to make a "classic" private messages extension. It can be as simple as adding a "Send me a message" button on user profile page and a popup with Subject, Body and a Submit button. Well, notifications are necessary, having threads would be nice. But I would keep private conversations one-on-one only. If people want to talk privately there are other means to do it. Forums are supposed to stay primarily for public conversations, but they do need to provide a way for people to write to each other directly occasionally.