Rivals... To be honest I don't see most of those as being that to each other.
Shanks and Buggy are not rivals.
Luffy and Zoro are not rivals.
Luffy and Blackbeard are both after the One Piece and Pirate King title but it's not like they're actively competing and trying to surpass the other, they're each doing their thing and will fight in the end. Can this even be called rivalry? If so, every pirate after the One Piece could be considered one too.
Zoro and Mihawk are rivals? Never felt that. Mihawk has always been superior, how can he be a rival to someone weaker? He's the goal, the master, he wants Zoro to surpass him.
Roger and Whitebeard often fought each other and could be called rivals, I guess.
I can think of Zoro and Sanji (though it's a different kind) which you didn't include for whatever reason.