Skyhawk 172/Cutlass 172
Flight Crew Instrument; Flight Crew Flight Instructor; Flight Crew Commercial
Flight Crew Last 90 Days 167; Flight Crew Total 445; Flight Crew Type 219
Situational Awareness; Communication Breakdown
Conflict Ground Conflict; Critical
Flight Crew Took Evasive Action
Procedure; Company Policy; Human Factors
I was flight instructing in Aircraft X. After our runup; we thoroughly cleared the area for other traffic; which for us involved a 360 degree turn in the runup area to scan for traffic; and proceeded to make our full radio call with intentions to takeoff runway XX before rolling onto the runway. We also performed a radio check and we able to verify other aircraft talking on the radio; none of which were a factor for our departure. ZZZ airport taxiway A4 was designed to transition aircraft onto runway XX at an angle so that they would have good visibility of other landing traffic; so when we do our checks left and right before crossing onto the runway we are very easily able to identify traffic on final or base. We did our checks left and right and saw no other aircraft coming in. We did not delay on our roll; taxiing right onto centerline and applying full throttle without pause. About 500 feet down the runway; we felt our airplane shake and saw another aircraft (Aircraft Y) fly overhead in the same direction; presumably doing a go-around to avoid crashing into us. We were almost at rotation speed and I felt that the safest thing to do would be to continue the takeoff and get out of the way; so we did; as I watched Aircraft Y peel off in a steep climbing turn to the left. While peeling off upwind the pilot made a radio call 'short left base 34' and proceeded to make a tight 360 degree turn and land halfway down the runway behind us. This was the first time he had used the radio; and I was able to later verify by listening to the archives.The skydive operation at ZZZ has a reputation for negligent practices when it pertains to traffic patterns. What they do all day every day is drop their divers off at 14;000 (10;000 AGL) directly above the airport and then proceed to divebomb into an incredibly steep downwind and base; and turn final just off the approach end of the runway. They descend 10;000 feet in half of a traffic pattern and anybody else who is in the pattern is going to be in danger of not seeing them descend upon them. In this situation; I believe what happened is that the aircraft was positioned so high and tight on his base; that he was out of our view. It doesn't help matters that he never made a single radio call on downwind; base; final; anything. We did make our own radio call that we were departing rwy XX well before crossing the hold short bars; there was at least a 15 second period where the pilot could have used his own radio to say 'Hey I am on final XX; please hold short'.I believe that it is very important that this be brought to somebody's attention; because it is not the first instance of this skydive operation putting their own and other's lives in danger. There cannot continue to be issues with them that are 'too small and too spaced out to report to the FAA' until somebody gets hurt eventually.
Pilot reported an aircraft over flew their aircraft while on departure roll.