It exists as an extension:
They are private discussions, not private message like phpBB.
It does require a bit of setup, but I imagine this is 10s of minutes to configure and test locally before deploying.
Make sure you configure the private discussions permission on the Admin Permissions tab to your needs;
- Create private discussions with users
- Create private discussions with groups
- Create private discussions with blockers
- Edit recipient users of private discussions
- Edit recipient groups of private discussions
This way everything is a discussion like everything in WordPress is a post (simpler notifications, permissions, etc).