If you look at the part where the girl and the blond haired guy get their picture taken, I think both arms go limp for a second as the puppeteer changes hands. He shook the hand of the guy that's done posing with his left hand, then when the girl and the guy walk up, she holds out her hand to his right hand. At this moment, it looks as if both hands go limp and the head kinda jiggles before the right arm goes "live" again.
I'd say that the puppeteer has the ability to switch hands as necessary. This would also aid in keeping him from getting too fatigues while operating that head (by being able to switch hands).
Just a theory. I would think if the head was operated remotely, then both of the performer's hands could be active at the same time.