Flight Instructor reported a NMAC while entering the downwind at a non-towered airport.

Narrative

During a visual approach into 47N Runway 07 from the NNW We made inbound calls informing traffic on the 47N CTAF that we were approaching from the NNW with distance reports at 10; 7 ; 5; and 3 miles. The call included that we were entering on the 45 for left downwind traffic for runway 07. As we turned right on the downwind approximately 1.5 miles from the airport; we saw another Aircraft Y approaching on a 180 degree opposite course offset to the right at the same altitude. The aircraft appeared to be transitioning 47N [area] on a west to east course. The aircraft then announced that it was entering the downwind and proceeded to make a 180 degree right turn in front of us at the same altitude announcing that they were downwind as they turned. We immediately slowed and began to divert to the right to initiate a 360 turn to avoid the traffic while announcing our position and theirs over CTAF. The other aircraft then started a right turn; and announced that they were now making a 360 turn on the downwind. We halted our turn; and proceeded on the downwind. This situation was a result of a non-standard pattern entry procedure by the other aircraft; and a near miss. Human performance factors included poor communication and use of non-standard/unsafe pattern entry and pattern operations maneuvers by the pilot of the other aircraft.