I think if you are seeing this for a url (/members/username/groups), it is only showing groups for that member. I think you need to have it go directly to:
/groups
Not:
/members/username/groups
Try adding /groups after your domain name and see if you can see the public groups.
I was having the same issue. When I set up my menu, I saw a section on the left that had a BuddyPress tab below the Page tab. Turns out the BuddyPress options map to the user that was logged in. I don’t think all themes show this because I have a test environment that does not show the BuddyPress tab. Check your url and make sure that you are going to domainname/groups/ (or whatever your page name is for the groups). My site was going to domainname/members/username/ when I clicked the Groups link in the menu.
