Only a very few officials of the Vatican, here as diplomats in Washington DC enjoy any diplomatic immunity. The priests at your local church have zero legal immunity for crimes they may commit.
I understand that, but the Vatican has the power to tell their bishops not to report crimes and there is little anyone can do about it, except complain.
The Vatican has seldom asserted these rights, they usually transfer the offending priest to another diocese.
this does not address the sickness of the priest involved or the implied lack of respect for the laws the nation in which the priest has his parish.