It depends on the company and how far they go in their background check. Usually even if you are not paid, they still need to gather your information for HR purposes, and in that case, they will ask you for your SSN, etc.

I'd suggest you to ask around, use your judgment to determine how much you can trust this company and the person hiring you. If you are close enough, see if they are willing to give you this chance to experience (because the accounting field is all about experiences...) If you are not too tight with finances and a chance strikes, take this chance so you can actually apply what you learn at school, even if you are not getting paid.

I really hope this works out for you. I've been dying to get some kind of experience...but I never succeeded. I get really worried sometimes, because even if the DREAM Act passes, I've already wasted so many years doing non-accounting related jobs. It'll be very difficult for me to get an entry-level job while everybody else my age are already somewhat of a senior-level accountant.