The only way I’ve found to display private forums on the main forum listing page is to use the Private Groups plugin. From what I can tell, it does not use the BuddyPress groups feature — you have to create new groups within the plugin settings.
The good news is that the plugin does exactly what I was looking for: It allows you to have private forums for a subset of members, and the forum is not visible on the main forums page to anyone who is not part of that subset.
Note that the plugin requires you to use different “Recent Activity” widgets, in order to hide private activity from other users. Those widgets are included with the plugin.
If there is a way to do this natively with BuddyPress groups, I’d like to hear it.
