Pilot reported NMAC with another aircraft in the airport traffic pattern.

Narrative

Both involved aircraft were rentals. Runway (RWY) XX was active. Approached airport from south practice area and called long left crosswind for RWY XX about 1 mile out. On crosswind I contacted departing Cessna aircraft (one involved) and announced a right 360 for safe spacing. Call acknowledged.On roll out on a close left crosswind for RWY XX again at pattern altitude and deviated slightly to the left over RWY XY numbers to clear another departing aircraft. Called left cross wind again; and then called left downwind for Runway XX.At mid field left downwind an overtaking Cessna suddenly yelled on frequency 'you almost hit me MAN!. Reckless MAN!' He was overtaking at my 8 o'clock and about 200 feet above and 200 feet left. The CFI then starting yelling;'Reckless operation MAN!' I deviated to the right as he overtook me. I never saw him and wondered what he was speaking about. I remained silent not wanting to tie up the frequency in a 'pissing' contest.Thereafter I followed him on base to final; and landing.After landing I identified myself on the radio; as I taxied behind him. I stated I was sorry if there was some mis-communication. I was on crosswind doing a 360 for departing aircraft. The CFI said yeah that was me MAN!. He started again on frequency; 'That was reckless operation. MAN!'I requested a call from the chief pilot of the FBO that rented both planes. From what I could tell the CFI (first departing aircraft) did a 4 mile departure while giving student instruction. I lost track of him due to sliding left to avoid another departing aircraft when I was on the rwy XX left crosswind; now over the numbers of opposite RWY XY.I don't know if he heard my radio call outs; but I was well within normal pattern altitude and distances from the active runway. Since he stated he saw my earlier 360 on crosswind he should have known I would be somewhere close to the airport; when he returned on his long downwind. I didn't hear his call outs; but I did hear another Cessna calling out a 45 entry to my downwind.