GA Flight Instructor with student reported an NMAC while performing touch and goes at SPZ non-towered airport.

Narrative

Near miss in the Silver Springs Airport traffic pattern. Us and another training aircraft were doing training touch and goes at the silver spring airport. We had reduced climb performance (altitude/heat) and thus were turning crosswind at 500' AGL as to not get too far away from the airport. However; we were still flying what some might consider a larger than normal traffic pattern. The other aircraft was also doing touch and goes in the pattern and was able to fly a tighter pattern; and therefore was starting to 'catch' us in the pattern. We made our calls on departure leg; crosswind and downwind and just before we turned downwind I heard them make a call for a 'crosswind departure northbound.' As we rolled out on downwind I was looking for them out the right side of the aircraft; to no avail I looked at ADS-B on my ipad and saw they were right on top of us; same altitude. I immediately took controls from my student; took evasive action; and only then did I see them turning away in the opposite direction. In the post-miss radio chatter the other aircraft claimed they had not heard our radio calls; though they definitely heard us after the fact. I believe that the leading causes of our near miss was us flying a slightly larger traffic pattern; both instructors being distracted by teaching students; and especially the other aircraft flying a non-standard departure and not visually clearing the non-standard departure. Notably; here in the Sierra mountains I see pilots get used to flying non-standard departures due to the terrain surrounding many of our non-towered airports. I believe this creates complacency in flying predictably in the non-towered environment. At silver springs there is plenty of room to do a straight out; or 45 departure off the departure leg. There was no reason to do a non-standard departure at this airport