In an Automatic subscription, everyone is initially subscribed but can unsubscribe. The subscription setting affects all users, not just students. You can make use of "Forced subscription" and remove the initially subscribed capability from the Teacher role, but I would not recommend this as it prevents students from unsubscribing.
I would recommend the following instead:
- create one group per student + staff combination;
- place each of these new groups in a grouping;
- create one forum for all users (rather than one per combination): under "Common module settings", set the Group mode to "Separate groups", and set the grouping to the one you created in step 2.
This works best when the staff members are non-editing teachers. Since Moodle 3.2 (I think) the Non-editing teacher role does not have the "Access all groups" capability. This means that they can only see the groups that they are a member of, and they can only see the forum posts and discussions for their own groups.
Hope this helps,
Andrew