The symptoms you describe indicate that the aircraft doesn't have a proper intercom, but that the two microphones and the two sets of earphones are strapped in parallel.
Rather than buy a different headset (assuming you can't borrow a compatible one from the club), I'd suggest buying an intercom unit. Mendelssohn sell 2-place and 4-place ones. Plug it into the aircraft sockets, and plug you and the instructor into it.