The ground training required by 61.31(g)(1) can be given by any qualified ground or flight instructor. The flight training required by 61.31(g)(2) would have to be given by a flight instructor with the appropriate category and class ratings on both his pilot and flight instructor certificates - 61.195(b).
If the aircraft used for the high altitude endorsement was multiengine, then the flight instructor would have to be an MEI.