2.1.2 If the only payment involved is the payment of the pilot, the flight is deemed to be private for airworthiness purposes (although it will still be aerial work for other purposes, e.g. flight crew licensing). This enables a private owner to pay a flying instructor for a flying lesson in his own aircraft even though the continuing airworthiness requirements that would be applicable to public transport aircraft may not have been applied.
I would expect the FAA to ensure that pilots giving any form of instruction in the USA meet the US rules and requirements....Just like the CAA would....unless possibly it is in a G reg aeroplane.