Hmmm, good question. They do ask if you're a US Citizen or permanent resident. But how they verify, I'm not sure.
A while back before I became a permanent resident, I applied for a mortgage and indicated that I was neither a US citizen nor a permanent resident. I was approved for the loan, but after the closing, I looked at the documents and it indicated that I was a US citizen (the documents were prepared by my real estate agent prior to the closing). Anyway, there was no hitch with the loan and the bank did not require a proof of citizenship.
So with student loans, who knows.