How do I show the members in a block, contextually filtered by the currently logged in user?
One custom solution that is doing the exact same thing in my case is to create a view-block and show group content of type group membership. Then add the relationship Group content User, require this relationship, filter by plugin Group membership. Then add a contextual filter Group content: parent group. Add all fields you want to show in the view and use the relationship Group content user to display name, id, email etc. 

Now the "custom" part in my solution is a tiny module that implements the hook_views_pre_view() and adds the gid of the currently logged in user to the $args array of that specific view like this: $args[0] = $gid;