Going a bit further afield...
US Federal Code Title 14 Part 375:
§ 375.24 Entry and clearance.
All U.S. entry and clearance requirements for aircraft, passengers, crews, baggage and cargo shall be followed.
Ok, so maybe not what you was after...
From your example, Pilot 1 gave PPR and so customs/immigration was notified. They may (according to the laws of the appropriate land) decided that no further action was necessary, but they knew about it so all's well - or they may have decided to show up to take a look. Pilot 2, however, didn't notify, so customs/immigration were not aware and so had no say in the matter.
I think it comes down to whatever the law of the land is concerning border protection, customs and immigration - the relevant AIP Gen 1.2 (I knew they were all the same for a reason
) should be able to tell you...