Not all rivalries are bitter. Some are actually quite friendly with both parties getting along very well. The rivalry usually manifests in a competitive drive to one up each other with both parties developing their talents. If the rivalry involves people of the opposite sex, there is always the possibility of them becoming more intimate with each other.
Can be related to Friendly Enemy, depending on the circumstances.
Anime and Manga
Bakuman。 is loaded with them. Just about every character is competing with one another, but you'd be surprised how well they get along.
Tycho Science and Princess Lal'c in Aim for the Top 2! Diebuster. As the top two ranked pilots, they constantly traded friendly jabs at kill counts.
Naruto and Sasuke (pre-time skip) as members of Team 7.
Sakura and Ino become this upon Sakura's discovery that Ino also has a crush on Sasuke.
Guy and Kakashi, as proclaimed (multiple times) by Guy. Kakashi, true to character, is rather apathetic about the whole thing.
Though only vaguely alluded to, the reader gets a sense that Jiraiya and Orochimaru had a relationship much like the one shared by Naruto and Sasuke.
Eyeshield 21: Sena has friendly rivalries with both Shinn and Panther; in the case of the latter the two of them are practically friends, despite being on opposite teams. The same can be said of Ootawara and Kurita, and Sakuraba and Monta; one gets the feeling that if they played at the same school they would be the best of friends. Quarterback Youchi Hiruma isn't particularly civil to anyone, but his rivalries with Ojou quarterback Takami and Seibu quarterback Kid is about as close as he gets. He and Hakushuu quarterback/safety Marco verge on Friendly Enemy.
An interesting case is Hiruma and Shien, who for some reason (considering Hirumas personality) get along splendidly. They both think the other one is the best quarterback in the region. Even in the final arc, Hiruma still trolls Agon by saying Shien is a better quarterback than him.
Harlem Beat: Kuwata from Yotsuya Unohara team has no qualms about teaching Naruse or conversing about team management with Kobayashi; both from rival team Johnan.
In Mahou Sensei Negima!, Negi has this relationship with Kotaro. Likewise, his father Nagi has this relationship with Jack Rakan.
In the manga incarnation of Higurashi, this is the way Mion thinks of Keiichi. She also has a major crush on him too.
Ichigo and Ishida of Bleach become this after some time.
Tomo and Kagura of Azumanga Daioh, both having learned that Sakaki doesn't particularly rival back — she just wins and gets on with her life — seem to settle into this by the end of the third season.
In Fairy Tail, Natsu's rival started out as Gray until their relationship gradually turned into this, while his former Evil Counterpart Gajeel became his new rival and he gained newEvil Counterparts in the short-lived Zancrow and Sting. Though Natsu may end up forming a friendly rivalry with both Gajeel and Sting, too.
Happy sometimes sees Lily as this, especially when it comes to Carla.
Though not without jerk rivals, Ash of Pokémon has several rivals he's friends with, most notably Bianca and Stephan. All of Dawn's rivals aside from Ursula are also friendly, and May has Solidad and Drew. (The latter after seeing May progress.)
This is how Spider-Man and The Human Torch's dynamic was for years. They become less competitive with each other as they get older, but there's still an element of ragging on each other/trying to outdo each other there, only now mixed with actual respect - to the extent that Johnny left instructions for Peter to replace him on the team in the event of his death.
Superman and Batman went through a lengthy phase of this in roughly the middle third of their friendship. These days, it's entirely down to the occasional joke between them - but the rivalry seems to have been handed down to Superboy and the third Robin.
In David Weber's The Honor of the Queen, relations between chancellor Prestvick and Protector Benjamin Mayhew were described as this concerning interpretation of of monarchical power limits.
Zach and Hodgins on Bones were like this. They both wanted to be honorary "King of the lab".
Inspector Gideon Pryke in the Jonathan Creek episodes "Black Canary" and "The Case of the Savant's Thumb". While his style is very different from Jonathan's, he's always happy to have someone else intelligent look at the case, and he and Jonathan are usually neck-and-neck in the deductions ... until Jonathan gets the Eureka Moment and Pryke misses it.
Many neighbor countries have some bitterness to each other, but are still major commercial partners - UK and France\Ireland, Brazil and Argentina, Japan and China\South Korea...
Despite his denials, Mao and Raspberyl of Disgaea 3 are friendly rivals.
A friendly rivalry between the protagonist and Jamie in Harvest Moon: Magical Melody is possible. Since Jaimie is always the opposite sex, the option of marrying him/her is always an option. Doing so lands you a rare Non Standard Game Over good ending.
This is apparently the relationship that Tails and Yoshi have in the Mario & Sonic series.
As well as with the main duo Mario and Sonic themselves.
And Peach and Amy; Daisy and Blaze; most of the rivalries in the games... However, it's averted with Luigi and Shadow, who spend most of the time on-screen glaring at each other.
Jack and the Blacker Baron in Anarchy Reigns are in constant competition with each other, and neither can resist chances to land verbal (and sometimes physical) jabs at the other's expense... but when actually pushed, the two will immediately band together to crush anything that threatens either.
Flavia and Basilio from Fire Emblem: Awakening rule the East and West halves of the militant nation of Regna Ferox. Flavia tends to call Basilio an "oaf" due to his grumpy attitude, and the two hold regular Combat by Champion competitions (usually won by Basilio) to determine the full ruler of Ferox, but they are quite friendly with each other, both on and off the battlefield.