GA Instructor Pilot reported an aborted takeoff when they noticed another aircraft was taking off opposite direction on the same runway.

Narrative

While on IFR training mission; we stopped at ZZZ to refuel. The weather was VMC with calm winds. After refueling; at or about XA38; we notified ZZZ traffic of our taxiing intentions 'ZZZ traffic; Aircraft X; taxiing from fuel area to Runway XX; ZZZ.' via CTAF radio.Upon reaching the end of the taxiway and before entering the runway; at or about XA40; we notified CTAF of our departure intentions: 'ZZZ traffic; Aircraft X; departing XX; ZZZ.'Both myself and my CFII checked visually left and right and did not see any traffic on the ground nor in the air. Nothing showed on ADS-B. I taxied the plane onto Runway XX and moved the throttle to full power. As we accelerated to rotation speed; both my CFII and I spotted a low wing plane heading south; directly towards us; about 50 feet above the runway. As I was not immediately sure if they were landing or taking off; I pulled power off; began braking; and moved to the right side of the runway.The plane was taking off on Runway XY but never spoke on CTAF until after he was leaving the pattern when he identified himself as an RV10; departing to the southeast. Neither I [or] my CFII had the presence of mind to write down his tail number.After I exited the runway; my CFII double-checked to make sure that we were tuned to the correct CTAF frequency (we were). We both are certain that the RV10 never spoke on CTAF until after he took off and began to leave the pattern. My CFI checked his Stratus several times and never saw him on ADS-B; either.We taxied to Runway XY and took off without incident. As we took off; it became clear that due to runway slope and 'dip' that the end of [Runway] XX could not been from [Runway] XY. We are not sure that the RV10 ever saw us.This incident highlights the absolute necessity for proper radio communications at all times; even at so-called uncontrolled airports.