I enabled OG and now I have a tab that lists all "public" groups if you're not registered, and it'll show the groups you belong to if registered. What's the best way to get rid of the groups tab for Anonymous users? It can be visible to authenticated users.
The quickest way I can think of doing it would be to use the Tab Tamer module. I've used it to do the exact same thing on a Drupal 6 site and it worked really well.
I had a similar problem with OG Extras installed, the view and members tab would show even when a site visitor wasn't logged in, I wanted it hidden for non registered users or non-admin users.
To stop it showing I edited the view 'OG Extras group members (User)' made sure I had the Page, not the block of the view selected and changed 'Access:' for the 'tab menu' to Role | administrator, this means that when logged in as an administrator I can see the desired tabs but when logged out the tabs are hidden.
Also if you created the menu links by the modules views, you can restrict access to the menu link and the viewed page from the views module.
OG Invite People provides an "invite" for existing users as well as new users. In 2.x it has an option to disable the OG group tab. Usual use case is to make your own "manage members" tab beneath via views or panels.