When they became rivals? Mihawk was already famous during Roger's era, Shanks made his fame from his duels with Mihawk.
Shanks at some point attacked Mihawk or the other way around.
Shanks at some point attacked Mihawk or the other way around.
Yup, either Shanks or Mihawk at one point in the time sought the other for a fight and then they ended up becoming rivals. Why do you think Shanks immediately thought Mihawk came to him for a duel as soon as he paid a visit? They were fighting to one-up each other. They have a reason for their duels. This is different from attacking enemies.