The pilot of the tailwheel-equipped Piper, reported that his airplane was not equipped with a radio, and that the forward visibility is not good when taxiing, while piloting the airplane from the rear seat. He entered a left downwind for the runway, saw no other aircraft while on final, and continued to land. He added that during the landing roll, as he turned to clear the runway, his airplane collided with a Cessna airplane that was taxiing on the runway after having landed on the same runway, from the opposite direction. He reported that he did not see the Cessna prior to the collision. The Piper sustained substantial damage to the right wing's front spar. The pilot of the Cessna reported that while flying local in the traffic pattern he announced his position and intent to land on the airport's Common traffic advisory frequency (CTAF) during all landings, while at the same time visually checking right and left for traffic. He added that as he was taxiing on the runway toward the exit after landing, he saw the Piper land in the opposite direction on the same runway. Subsequently, the Piper turned diagonally across the runway toward the Cessna, and the two airplanes collided. The Cessna sustained substantial damage to its right wing. Both pilots reported that there were no pre-accident mechanical failures or malfunctions with either airplane that would have precluded normal operation.