Small aircraft pilot reported an NMAC with another small aircraft at a non-towered airport on take-off. Pilot initiated a right turn to avoid a collision.

Narrative

The other aircraft involved was an Aircraft Y.BWG; winds were [email protected] student and I had were back taxiing on 30 for departure. We heard Aircraft Y on the CTAF saying he was 'taxiing to Runway 21; for a downwind departure to the north.' This would be the last transmission we heard from that pilot. I immediately transmitted over CTAF that we were 'backtaxiing on Runway 30 and would be a crosswind departure to the south.'I thought the other pilot's runway selection was a strange course; choosing the longer runway; but with a crosswind gusting to 20kts.Less than a minute later; my student and I had completed our back taxiing and were lined up in position for departure on 30. We were simulating a short field departure; so we held in place for a moment as RPM stabilized at full power. We couldn't see Aircraft Y anywhere. Not by sight; and not by ADS-B. We let off the brakes and carried out a textbook short-field takeoff. Not long after we left ground effect; Aircraft Y came into view just to our right and on their climbout from Runway 21. He was much faster than our aircraft. I took the flight controls and turned right to avoid a mid-air collision. The closest we got to each other was probably within 400 feet horizontally and 150 feet vertically. Once clear of conflict I transferred controls back to my student; who continued our takeoff and departed the pattern to the south off the crosswind leg. We never heard from the pilot again. He did depart off the downwind. I tried to find his tail number on ADS-B just moments after we were above 1500' AGL. No luck. I suspect he turned off his ADS-B because he knew he'd screwed up. I asked my student 'Did I miss anything? Did you hear him say he was taking off; departing; on the roll; or on the go for Runway 21?' My student didn't hear anything of the sort either.To the best of our knowledge; this guy made a single radio call to say that he was leaving the ramp area and taxiing to Runway 21 via Taxiway A. Never made an announcement that he was turning onto the runway or beginning his takeoff. We were very clear in stating our every position; action; and intentions.