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.
My recollection, which is not as reliable as it used to be, is that "staffs" is always plural for "staff" in British English (two staff members are two staffs), and most American English speakers use "staff" for both singular and plural. However, I sometimes hear the word "staffers" coming out of WDC news stories. The only time I seem to hear "staffs" as plural of "staff" is in British English.