C172 flight instructor reported while on departure during a touch and go another aircraft without transmitting on the CTAF departed a crossing runway resulting in a flyover of the other aircraft.

Narrative

I am a flight instructor and was performing private pilot instruction in a CE-172. I was providing my primary student instruction on airport traffic patterns. We were flying the left-hand pattern to RWY XX and were practicing touch-and-go's on RWY XX at ZZZ. My student was announcing position on each leg of the pattern including a 'Final RWY XX; touch-and-go' radio call. As we were rotating; another aircraft (Aircraft Y) pulled onto a crossing runway; RWY XY without making any radio calls. We flew over the top of Aircraft Y by less than 100FT. By the time I saw this aircraft entering the crossing runway; should we have had a power plant failure or needed to abort for any reason; this would have ended catastrophically. Aircraft Y departed RWY XY and then made an immediate westbound turn without making any calls. I attempted several times to raise Aircraft Y on CTAF and guard with no response.Aircraft Y pulled onto active Runway X at ZZZ with and aircraft already rolling for departure on the runway. I visually saw Aircraft Y violate the runway protection area as we were rotating; my student continued rotating and went airborne; there would not have been time to stop. My student continued and we flew over the top of Aircraft Y at approximately 100ft. The pilot of Aircraft Y continued slowly rolling across RWY XX on RWY XY runway so I don't know if he ever saw us. Cause: NORDO aircraft; aircraft using a conflicting runway with traffic already in the pattern; pilot-controlled airport.