Good news, everyone: private messages are back! Just hover over a user's avatar, then click on the three-dots-button and select "Send XYZ a message".
It works a little different than what we were used to in the old forum, as it's not so much a private messaging system as a "private thread" system. So when you write a PM, you need to pick the category (tags) under which the PM will be listed. There's no dedicated list of private messages, but they act just like normal threads that are visible just to the participants.
That system has its advantages, but it takes a little to get used to. The main downside right now is that you won't get an email if there's a new message. However, it's a start, I guess. What do you think?
In any case, the advantage of "Flarum" is that it's a live and thriving ecosystem that's constantly being worked on. So we can expect enhancements of the PM Extension in the future, or maybe somebody will even develop an alternative that is more similar to the old system.