Users considered staff only for Admins?

I've added all staff to my website but can't select them in the tickets. All have above Subscriber rights but none Admin like I do. This is important because the staff reply to Billing issues, Customer Service issues, and Product Support issues while I reply to Website issues and Server issues.

It looks like I have to give my staff Admin privs just to allow them to have assigned tickets. Is that right or am I missing something? It lets me customize like crazy who can create tickets, who can view tickets but I don't see any way to customize who can work tickets.