Pilot reported a NMAC with an aircraft on downwind at a non-towered airport.

Narrative

I was departing Runway 28 at GXY after full stop landing. Took off and announced 'Taking off Runway 28 remain in pattern right traffic for 28'. There was traffic in downwind leg for 28 also. Then another pilot announced 'Aircraft Y 2 miles west of field have the departing traffic and downwind traffic in sight; will enter right downwind for 28'. Upon climb out after starting cross wind we had near miss; aircraft was below and passing in front of us about 300 ft horizontally and 200 ft below; the 'Aircraft Y' We had ADSB IN in cabin and had downwind traffic on screen and visual however Aircraft Y did not have ADSB and was not shown on screens. After near miss Aircraft Y called 'I just had traffic coming right at me' I reported yes that was me; I did not hear your position report that you had entered downwind and did not have visual' he reported 'We will follow you in right traffic for 28' I said 'No go ahead; (as he was now already ahead of me in pattern) we will make a left 360 for spacing and follow you'.Pilot continued to report as 'Aircraft Y' And I did not get a tail number. Better position reports could have been made by both parties to understand where we were at in relation to one another; ADSB also could have helped prevent this near miss if it was required in all aircraft.