I’ve had a look at setting the notification level for group messages when the group’s “Who can see this group’s members?” setting is set to Group owners (or Group owners and staff.) When a group is configured in this way, there is no inbox displayed for the group on the messages tab for members who are not group owners.
When I test this on my development site, the other thing that stood out was that the member who is not an owner receives notifications about group messages, but the message is not displayed in the user’s inbox. The message exists in the inbox, but no link is displayed to access it. The user can access the group’s message from their inbox by going to /u/<username>/messages/group/<group_name>, but no link to that path is displayed for them. The user can also see the group’s messages from the group’s Messages tab.
If the views the PM and clicks on its PM icon, they will be taken to /u/<username>/messages/group/<group_name>. On that page they can set their notification level for each group PM, but can not set a default status for all PMs sent from the group:
There is an issue with the notification button in the screenshot. Unless someone knows what is causing the issue, I’ll test all this again on a production site.