I just discovered this was flarum private messaging! (because of the name, I didn't realized it).
It works great, but I'm a little unsettled too by the way it's really mixed with other discussion. One first easy step to separate it, at least symbolically, could be to propose in the options panel to just make it an icon in the top right (near the bell), rather than an entry in the menu like other forums?
There could also be an option to remove those discussions from the "all discussions" section.