C152 instructor pilot reported an engine failure on departure. The pilot immediately returned to the airport and landed on the right side of the runway to avoid an aircraft that Tower instructed to move to the left side of the runway.

Narrative

Student pilot completed preflight satisfactorily; engine start; taxi and run up complete under instructor pilot supervision. Run up was within limits. Ended the Emergency Brief by instructor pilot audibly saying: 'In the event of an actual emergency; I will take controls and coms.' Verified fuel amount; fuel selector and fuel quantity audibly. Briefed the take-off. Student pilot would have controls for a soft field take-off. Called Tower for a northwest departure holding short of XXL. Briefed the take-off distance when we entered runway. Student pilot slowly applied full power with no brakes; rotated before Vr (soft field); stayed in ground effect and got to Vy to start climb out. After 50 ft.; retracted flaps. Student pilot climbed up to 1200 ft. before Tower requested us to start our turnout NW bound. Student pilot started turn to the right. Aircraft experienced engine failure (still full throttle but RPMs decreased to idling). Student pilot audibly said 'your controls' to which instructor responded 'my controls' and [requested priority handling] while applying carburetor heat and full power. Instructor pilot saw runways to the right and decided that the aircraft was capable of making either runways. Instructor pilot chose Runway XYL; but upon completion of turn; Instructor saw aircraft on the runway and notified Tower that they were sidestepping to Runway XZR. An aircraft was on that runway as well; but Tower notified us to use the right side of XZR and that the opposite aircraft would move to the left part of XZR.Instructor forward slipped the aircraft for a couple seconds before deciding that the aircraft would not overshoot the runway. Instructor landed aircraft safely with engine still idling. Instructor pilot taxied to and contacted Ground after clearing the runway. Engine still was idling around 500 (throttle out)/ 1300 (throttle full). Ground cleared aircraft to taxi and asked instructor to contact Tower by phone number once aircraft was at the ramp. Instructor completed engine shutdown and checked for mags to be grounded which they were. Instructor contacted Tower by phone once the aircraft was secure and parked. Tower was wondering about the N number of aircraft; due to instructor saying incorrect N number upon [requesting priority handling]. Instructor explained that he was focused on flying the aircraft (Aviate Navigate Communicate) and apologized for the confusion with the tail number. Tower told instructor that it wasn't an issue; but that they just needed to confirm the aircraft and get the credentials of instructor (phone number; and name). After debrief it was determined that the intake valve had failed and broken off causing misfiring and then engine to fail. The aircraft was 2.6 hours away from a 50 hour inspection per the school standards. The aircraft did not have any other issues between the previous 100 hour and the flight.