It would be difficult to describe Facebook as private under any circumstances, whether in Saudi Arabia or not. Obviously, anything you post on Facebook should be considered public.
Additionally, Facebook will hand over data in and about accounts in Saudi Arabia in response to a valid legal request.
According to Facebook, the KSA government filed three requests for data since 2013 and Facebook complied with one of them. The KSA also filed six “preservation requests” for a total a 10 accounts. When a preservation request is made, according to Facebook, “[…] we will preserve a temporary snapshot of the relevant account information but will not disclose any of the preserved records unless and until we receive formal and valid legal process.”