I’ve tried it with a private forum in my development environment, both with and without the MemberPress plugin enabled and synced logout is working. I’ve tried it with a private forum on a production site and it’s working there too. I haven’t tried it with the MemberPress plugin enabled in production.
One thing to note is that on a private forum with SSO enabled, you can’t log out of Discourse when you are logged into WordPress. Discourse automatically logs you back in. That probably doesn’t have anything to do with what’s going on here though.
One thing that I found with the MemberPress plugin that is quite concerning is that users are able to login to Discourse without first confirming their email address. When using SSO through the wp-discourse plugin, it should only be possible to login directly to Discourse without an initial email confirmation when users are registered through the normal WordPress registration process. This is the first plugin I’ve tried that is not allowing the wp-discourse email verification function to be called. I think there’s a change I can make to the code so that it will work properly in all cases.