Flight Instructor on downwind in a VFR pattern reported a NMAC with another aircraft which did not follow Tower's instructions.

Narrative

We took off from Gateway and requested 2100 ft. short approaches followed by a Johnson Ranch departure to the practice area. The event occurred after the first touch and go. We were told to make left closed traffic for Runway 12L. We were in the early phase of the downwind leg probably abeam the opposite runway 1;000 ft. marker (30R) and we see a plane on our Multi Function Display (MFD) getting closer to us and climbing towards our altitude of 2;400 ft. We didn't break out of the pattern but myself; the instructor; took controls and descended us down to about 2;100 ft. During this descent Gateway Tower identifies the plane as Aircraft Y and says 'Aircraft Y make a left 360 now; you were told to fly straight out and turn crosswind behind the plane; ahead and to your left. that traffic is on your right and that was very dangerous; you almost turned on top of them' Aircraft Y replied with 'no you said to look for traffic on the left; this traffic is on our right' Gateway Tower said 'they're on your right NOW; they were on your left when you were instructed to fly straight out'.Aircraft Y continued their 360 and reentered the pattern behind us; we finished the touch and go and requested to depart East (instead of Johnson Ranch) to get out of the airspace. The MFD reported the plane was less than 100 ft. to our altitude and they were approximately 200 ft. horizontal from us.