Training being conducted by a certificate holder would clearly be fair game, including 121, 135, 141, 142, (91K?), etc.
But if you were doing a part-91 training flight (or sim session) I think the fed would be limited to either a ramp check (airplane) or inspecting the sim's certificate. He would not have the authority to stand over your shoulder or invite himself along on a training flight if you didn't want to have him there.
But it's common for feds to be heavily involved in training if it's a new certificate, new training program, or a new fed. Just your bad luck to be a student in that situation...personally I avoid start-up training programs until they get the bugs worked out.