An FAA license holder flying an N-reg plane outside the USA, needs his personal radio license. This is obtainable from the FCC website for about $100 or something like that, and lasts about 10 years, IIRC.
Whether an FAA PPL holder flying a G-reg plane (which he can do, worldwide, VFR, ref ANO Article 26) needs the FAA radio license, I don't know. Probably does....
The N-reg aircraft needs another radio license; this is normally supplied (for a fee) by the US Trustee who owns it when based over here.
EU flying has a lot more red tape than American flying, but it isn't a big deal. One develops a process whereby one checks, pre-flight, stuff like PPR, Customs, etc all in one go. Obviously anybody flying just within the USA (which is what about 99.9% of US pilots do) doesn't need Customs, US airports are not PPR, doesn't need to check operating hours (most if not all US airports are open 24/7), etc. But he needs to get a briefing (weather, notams) which can be done with a phone call but which most "modern" US pilots get over the internet, and IIRC he needs a flight plan for IFR.
The "European red tape" can be a problem because actually flying somewhere useful is not taught in the PPL
But one can pick it up by asking on the internet (like here
).