This is where my "Train Wreck" theory tends to bear some truth. The Privacy settings are all over the place and I'm not sure that's entirely necessary.
1. Look in Admin/Settings/Privacy Settings. It would seem that each of the modules and the profile can be made private here. I made Photos private and I'm not sure what it actually did. In the absence of a manual, you'll have to experiment with that one.
I also switched Profile to Members only and the profile button disappeared from the main menu. That was a good sign. It also removed a few other buttons, but I'm not sure which.
I then made Events Member's only but the button also stayed in the menu and it works to a degree. There's a "Join" icon where the Albums icon usually is, but if any photos appear they're live.
2. The next derailed carriage is your profile. In every box there's a yellow key. You may have to go through all the boxes and change the Public key to a Members Only key.
3.Then there's the Albums. Each album can be made Private, members or public. That's easily done when you create a new album, but quite a pain when changing the setting for an existing one.
4. Finally the Caboose! Got to Settings/Dashboard (Initial Settings) Item 8 Permissions. Click on Non Member and disable all the actions that may be causing problems.
On another site I use, all these items are readily accessible by Admin, but most importantly they have a single setting - Public or Membership site. Just choose the one you want and the public is faced with a walled garden. (Nerdy term). I thought Dolphin had the same, but it's so hard to remember where to find things in the chaos.