I'm not 100% of all legalese technicalities but I know for a fact vatican law applies there and it's managed and guarded by vatican's gendarmerie (my ex went to the university behind the basilica which is also part of the Holy See).
They are like foreign embassies, part of Italy but under ownership of Holy See. The Holy See is different than Vatican City. The Vatican City is independent country governed by the Holy See, the head of the Catholic Church.
