I found that non-site-admins cannot edit any of the custom fields attached to their profile. I have found the Organic Groups (fixed) feature request "Authenticated users cannot edit his custom profile fields" that appears to explain the issue, but that's not clear enough for me.
Who exactly is the Group Audience for the User (the User?) and how do I set up the settings correctly to ensure all users can access and view their custom profile fields?
Is the User a "Group" and the Custom Profile Fields "Group Content"?