I am a teacher and we are in doubt if the plural form of staff is staffs. We have been debating on this topic.
"Staff" as a collective noun meaning a body of people takes either a singular or plural verb depends on whether you're thinking of the body as a unit:
The staff is unanimously opposed to the dean's recommendation.
or as multiple individuals
All the staff were shocked by the dean's recommendation.
The plural of "staff" is "staffs," and when you talk about staffs, you're talking about separate bodies of people, which, of course, requires a plural verb:
The staffs of both colleges were supportive of the joint program.