Flight Instructor reported another aircraft entered the traffic pattern against traffic flow. The other aircraft did not have all required external lights working and landed while the instructor was still exiting the same runway.

Narrative

I was performing landing training with a student. On the upwind leg of the landing pattern there was a garbled call on the radio which sounded like somebody asking about traffic at ZZZ. I asked calling aircraft to repeat. The aircraft came back and asked if there was any traffic at ZZZ. I responded 'one aircraft in the traffic pattern currently on upwind'. My student and I saw the traffic at our 2 o'clock traveling in the opposite direction to us; parallel to Runway XX which we had just taken off from. We continued to fly our pattern; left downwind followed by a left base. We made our traffic calls on downwind and base. When we were on base we heard a call saying Aircraft Y base to final. We looked around and could not see another aircraft; so proceeded with caution. We turned final and called the final; still looking for the traffic but did not see anything. We landed and at the end of our landing roll we saw a Cessna exiting the runway. The Cessna did not have a functioning rotating bacon or white tail navigation light. The green navigation light was functioning. The pilot of the other aircraft got on the radio and said 'so you landed while we were still on the runway'. I responded with 'Do you know that your beacon and your white light are out; we could not see you'. The other pilot again said 'you landed while we were no the runway. I asked 'did you hear that your lights are not working; we did not see you and you were not making radio calls?' The other pilot follow-ed up with a rude response so I did not engage him any further. I wanted to tell him that ZZZ has a left pattern to Runway XX and he had flow what seemed to be a right pattern; without a functioning beacon or white light and not making radio calls.