I’m going to need to see a few screenshots…
1. The Connections : Settings page, General tab.
2. The Connections : Roles page.
3. The Edit page of a Connections entry of an entry that should be visible.
4. The User Edit Profile page of a user who can not see the entry.
5. The Edit Page of the page that contains the Connections shortcode.
Those screenshots should clue me in on what the issue might be.