Single pilot with instructor reported an NMAC with a non-communicating aircraft at a non-towered airport.

Narrative

Performing touch and go landings and takeoffs at ZZZ airport on runway XX for tailwheel endorsement training. I was pilot flying and receiving the instruction; seated in the front seat. Our aircraft is a 1965 100HP Citabria with no ADSB out or transponder. After about two landings; a second aircraft; a Cessna 172; joins the pattern behind us. They are gaining on us in the pattern. We turn final; make a three point landing. Around; this time; the other aircraft reports final on the radio. We begin takeoff roll. On climb from the touch and go; I notice a shadow at top of windscreen. I glace at Foreflight with ADSB-In from a Sentry unit and see traffic hit right on top of ownship. I look back up at the shadow and it has now quickly become the belly of an airplane - I estimate well within 40-60 feet and closing. Oil stains; etc. clearly visible. I immediately pitch down; then bank slightly right. The other aircraft cannot see us. They had performed a go-around and had begun to overtake us on the runway during our takeoff roll; staying over the runway when they should have sidestepped to the right and maintained visual separation with us. They made no radio calls for a go-around. About 1 second after I pitch down; my instructor attempts to take control of the aircraft and pitches us back up. I say something to the effect of 'my controls; there's a plane right above us' and steer us to the right of the extended centerline. (Instructor later said he thought we had a flight control issue. He could not see the other plane at all. I did not immediately announce that I saw the plane as I was startled by it.) We got stabilized and rejoined the pattern behind the Cessna without additional incident. The Cessna performed one more touch and go and departed the area to the East. Conditions were daylight; SKC; excellent visibility; and calm winds. We were the only two planes in the area at the time. Several people on the field claimed to have seen the incident. I believe the incident could have been easily avoided had the other aircraft sidestepped to the right during the go-around; rather than assuming we were a full stop and overflying us. Additionally; the CTAF frequency was extremely busy this morning; preventing/blocking some transmissions.