Cessna pilot reported that while departing an uncontrolled airport; encountering an opposite direction aircraft that flew underneath and to the right of their aircraft resulting in the opposite direction aircraft. The pilot took evasive action to avoid a collision.

Narrative

We were taking off of RWY XX out of ZZZ. While we were getting ready to takeoff out of ZZZ at XA:40; Aircraft Y was making radio calls that they were maneuvering in the area to do a circling maneuver from either XY/XZ circling to XX. We checked the area and made sure the approach and departure ends of the runway were clear before we taxied on the runway to takeoff. Prior to taking off; we made a radio call stating that we were taking off of runway XX in ZZZ. Somewhere before 1000 ft; Aircraft Y passed under/to the right of us during the takeoff climb. We had no idea that they were flying right into the departure end of runway XX; which was the active runway at the time. I made a radio call to the pilot of Aircraft Y notifying them that they nearly hit us when we were on the climb out. They then told us that they had been making radio calls; and that they had us in view the whole time. After that remark I never said anything back to them. Aircraft Y should not have been flying into the departing traffic of ZZZ; what they did was unsafe and will cause an accident.