SR22 pilot entering downwind to a non-towered airport reported an NMAC with another aircraft in the pattern.

Narrative

I was initially approaching ZZZ straight in to Runway XX. Weather reported calm winds and [Runway] XX is the calm wind runway. Following my initial radio call; someone responded that the winds were favoring [Runway] XY. I waited for any reports from pilots in the pattern. There were none. I radioed that we would switch to Runway XY and do a 45 entry into left downwind. As we entered the 45; another pilot announced he was base for Runway XX. Since Runway XX has right traffic; this was concerning. I looked at the base leg and did not see him. My passenger saw him at 12 o'clock; 75 feet above us. I made a hard right turn and narrowly avoided him. He radioed 'did you see me?' To which I replied 'yes I did.' I radioed that I would extend my downwind to allow him time to land. When I heard nothing; I announced my base turn and he replied he was clear of XX. I landed safely. ZZZ hosts numerous experimental (he was one) and gyrocopters. For whatever reason; many of these pilots do not make regular radio calls. Since [Runway] XX is right traffic and [Runway] XY is left traffic; the downwind legs are coincident. To improve safety; the airport should reinforce the use of [Runway] XX when winds are calm and encourage better radio communication.