Flight Instructor on short final approach at a non-towered airport reported a near collision with an aircraft on approach to the opposite runway.

Narrative

Myself and my student were conducting a pattern work lesson at ZZZ. Ourselves and a Cherokee were in the pattern prior to the arrival of the other aircraft involved in the near collision. We were using Runway XX. On our upwind leg I noticed another aircraft approximately 1-2 NM south of the field over the river at approximately 1000-1300 feet MSL flying north/northwest. This aircraft did not have ADSB-Out and was not making radio calls. I made a radio call alerting traffic in the area of the aircraft. When we turned into our downwind I lost sight of the aircraft. Right before I lost sight I thought I saw the aircraft turning towards the south. During this time the Cherokee that had also been flying in the pattern had taken off from Runway XX and turned into a crosswind. I kept looking for the aircraft not making calls while we continued our pattern and approach to Runway XX. On short final for Runway XX the Cherokee; who was on a downwind for Runway XX; made a radio call alerting us that there was an aircraft on short final on the opposite runway. I spotted the aircraft; took the controls from my student; and initiated a go-round making a steep climbing turn to the right. The other aircraft around the same time also initiated a go-around. I re-entered the pattern for Runway XX keeping an eye on the other aircraft who then left the pattern towards the southeast.The core cause of this problem was lack of communication and situational awareness at a non-towered airport. The prevention of an accident is a direct result of good communication and awareness of the Cherokee and my action taken of performing the go-around. This situation could have been avoided if the other aircraft was also utilizing radio communications. The other aircraft involved in this incident is unknown at this time.