For US (N-reg aircraft), this is defined in 14 CFR 91.703, Operations of civil aircraft of U.S. registry outside of the United States. It says:
(a) Each person operating a civil aircraft of U.S. registry outside of
the United States shall—
(1) When over the high seas, comply with annex 2 (Rules of the Air) to
the Convention on International Civil Aviation and with §§91.117(c),
91.127, 91.129, and 91.131;
(2) When within a foreign country, comply with the regulations
relating to the flight and maneuver of aircraft there in force;
(3) Except for §§91.117(a), 91.307(b), 91.309, 91.323, and 91.711,
comply with this part so far as it is not inconsistent with applicable
regulations of the foreign country where the aircraft is operated or
annex 2 of the Convention on International Civil Aviation; [...]
That suggests that in most cases the pilot should follow the foreign (Italian) regulations:
- 91.703(a)(2) says you have to follow foreign regulations on "flight and maneuver"
- 703(a)(3) says you should (mostly) follow part 91 unless it's "inconsistent" with the foreign regulations
It all seems a bit vague to me, and what "inconsistent" means is probably the kind of thing that keeps lawyers busy. But also note that there are specific rules applying to operating foreign aircraft in the US. 14 CFR 91.711, Special rules for foreign civil aircraft, begins:
(a) General. In addition to the other applicable regulations of this
part, each person operating a foreign civil aircraft within the United
States shall comply with this section.
So foreign aircraft in the US are expected to follow US regulations as far as possible, which is consistent with 91.703: an N-reg aircraft abroad follows the foreign rules as far as possible, and a foreign aircraft in the US follows US rules as far as possible.
Since both the US and Italy are ICAO members, I would expect Italian law to roughly mirror US law on this, although that may be a dangerous assumption. For an authoritative answer, I would ask both the FAA and the Italian authorities.