The airplane was being used for multiengine instruction. The instructor said they were making a touch and go landing. He said the landing was normal and the student cleaned up the airplane. The student applied full power for takeoff and the airplane yawed left. The student overcorrected and the airplane yawed extremely to the right. The instructor took control of the airplane and attempted to get directional control but said the rudder was not effective enough. There were no brakes on the right side of the airplane. The instructor said he had reduced the power to idle. The airplane left the right side of the runway, struck trees and a fence and fence post, and came to a stop approximately 300 yards off the side of the runway, in a shallow depression. Later the passenger in the rear seat did state the instructor had reached for the throttles but only closed the right engine throttle, and the left engine ceased operation after impact with a fence post. No pre-impact problems could be found with the airplane or powerplants.