Flight instructor with student reported an NMAC while doing pattern work at RMY non-towered airport. Reportedly; proper radio calls were not being made which contributed to the event.

Narrative

While doing pattern work training at RMY Runway 28 in Aircraft X we came into close conflict with Aircraft Y. My student was making all traffic calls and the time in question he had promptly called base and final legs on the UNICOM. As we touched down on the runway another Aircraft Z called that they were crossing midfield for a left downwind RWY 28. Aircraft Y aforementioned then made their first radio call and stated they would enter the left crosswind. I quickly called to both aircraft that we were on the upwind and staying in the pattern. Aircraft Y was on a course that intercepted ours and made no attempt to climb or change heading. We at approximately 1400 MSL maintained altitude as he was proceeding from our right side directly over top of us. Neither aircraft made 10; 5; or 2 mile calls and we had called every leg of the traffic pattern. Had they given us any situational awareness that they would be entering the traffic pattern we would have either separated ourselves or been full stop to land until they were on the ground. As we were scanning for traffic we were able to avoid a mid-air collision that the other pilot did nothing including climbing to avoid that hazard. Vertical separation was 300-500 ft. lateral separation was 0.