I read that the Lateran Treaty (which created the Vatican City State) specifically allows Italian police to patrol St. Peter's Square, even though the square is within Vatican City. Practically, what happens when an Italian police officer arrests someone (say a pickpocket, probably the most common scenario)?
Does the policeman just unceremoniously take the suspect across the international border into Italy and to the police station with no special process whatsoever or does he have to first take the suspect to the Vatican City police station to have him formally turned over to Italian custody (i.e. "extradited")?
It seems to me that if there is no special process, even the "rubber stamp" kind, it kind of undermines the national sovereignty of Vatican City over that part of its territory. The Italian police would just be treating it the same as if it's in Italy.