It really depends on context. A phone rings on your desk and your friend picks it up. He talks for five minutes and you finally say "Hey! Who is on the other end of the line?". You go fishing with some friends. At some point you hook a fish and start a 30 minute battle. During the battle, a friends says "Boy Nozturk!! I can't wait to see what is at the other end of that line.". In this example, "on" could be used as well. You go to the movies and walk right up to the ticket counter without realizing there is a long line. Someone yells "Buddy! Your spot is at the other end of the line!". Again, on could be used. I could go on and on. Both expressions are used.

Both are correct