The paper itself reports about 10% of the outstanding amount (p.840) are private. However the data is pretty hard to come by that doesn't aggregate everything.

The paper itself reports about 10% of the outstanding amount (p.840) are private. However the data is pretty hard to come by that doesn't aggregate everything. > The majority of student loans — more than $1.4 trillion — are guaranteed by the federal government — up to 97% of the principal on the loan plus interest payments — in the event that the student loan borrower defaults. This guarantee ensures that most, if not all, of the underlying collateral will be returned to the lender in the event of a borrower’s default. That said, private loans do not come with the same guarantee, but these loans represent a small proportion — around 10% — of the student loan sector. The underlying presumption that student loans are federally guaranteed—because the overwhelming majority of loans are federally guaranteed—gives rise to the misconception that most loans included in SLABS come with federal backing; however, the reality is that they do not.