Good question SoCal as far as I know the instructor would have to be PIC in some cases such as the sea plane rating for the simple reason the student is not qualified to fly the airplane because they do not have the rating.
Myself I make it clear that I am PIC so there is no misunderstanding about who is responsible for the safe operation of the airplane.
I do know that when instructing on airplanes registered in Australia and Europe I had to hold an authorization from both C.A.S.A. and the J.A.A. to fly their airplanes and teach on same. I spent the last ten or so years of my career doing exactly that and as well I was issued an Airdisplay Authorization with no restrictions based on my Canadian license. Mind you I had to jump through all the same hoops as the rest of the air show pilots including an annual flight test.
The FAA would not give me the authorization to teach on U.S. registered airplanes without a FAA CFI, but did allow me to do type rating training and once the student was competent on type a FAA licensed pilot did a ride with them and the type rating was approved...weird huh?
Yeh come to think about it I did several type ratings in South Africa on a U.S. registered airplane and the S.A. C.A.A. issued the type ratings..go figure.
