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, or at the very least some UST.
Can be related to Friendly Enemy, depending on the circumstances.
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Anime and Manga
Bakuman。 is loaded with them. 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.
The two main characters Gon and Killua have this relationship to the max! Being the only kids in the Hunter Exam they instantly become the best of friendly rivals, and in the 2011 anime they even have a friendly race to the finish line.
Although Killua did have his moments of bitterness in the final stage of the Hunter Exam towards Gon that could say otherwise, but in the end he does outright say he just wants to be friends with Gon.
Meruem and Komugi also share this relationship when it comes to Gungi
Naruto and Sasuke (pre-time skip) as members of Team 7. Not the friendliest example, but they have their moments.
Sakura and Ino become this upon Sakura's discovery that Ino also has a crush on Sasuke. At first they'd actually drifted away from the "friendly" part, but regain it after their match in the Chunin Exams.
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 Shin 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 Youichi 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 Kid, who for some reason (considering Hiruma's 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 Kid 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.)
In Kin-iro Mosaic, there is Alice to both Karen and Karasuma in regarding to Shinobu (since Shinbou admires Karasuma and Karen really admires Shinbou). Regardless, she maintains a good relationship between the said two characters and any negative feelings are often Played for Laughs.
Variable Geo: Yuka and her best friend, Satomi, have known each other since childhood and have been competing against each other for just as long. The series even concludes with the opening exchange between them, during the final match.
In Tiger & Bunny, despite the competitive nature of the hero show and that the heroes themselves acknowledge each other as rivals, they remain friendly with each other, provide support and advice when the need arises and even throw a surprise birthday party for one of their own.
In Dragon Ball, Goku was this for many characters (eg Krillin, Master Roshi) early on before completely outclassing them, and had a more aggressive rivalry with some characters (Tien, Vegeta).
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 down to the occasional joke between them - but the rivalry seems to have been handed down to Superboy and the third Robin.
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.
Moze and Suzie from Neds Declassified School Survival Guide, once they stop being bitter enemies. They're friends, but still fiercely competitive. Best exemplified during the episode where they attempt to join a school club together but turn everything into an intense contest.
Moze: I don't think we're supposed to turn everything into a death match.
Suzie: Yeah, I know. We've been kicked out of every club.
In Tennis, the most famous example would probably be John McEnroe and Bjorn Borg. Some other examples are Chris Evert and Martina Navratilova, and Roger Federer and Rafael Nadal.
In The Beautiful Game, the rivalry between Japan and Australia has been made up of constant ribbing and fierce games, with both teams tied for wins as of writing. Japan was a prominent supporter of Australia's move to the EAFF Cup, and players happily perform in each other's leagues.
In Pokémon X and Y, your main rival (either Calem or Serena, who are also the options for the player character) is a friend whom you occasionally train against, sometimes you even fight tag-team battles alongside them.
In Soul Nomad & the World Eaters, Danette is Revya's childhood friend/rival. In her ending (male), it is heavily implied that they will become intimate.
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 Jamie 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.
Though it started off on a bitter note, Sol and Ky's rivalry eventually becomes more brotherly. By the time of Overture, they're practically a combination of this, and Vitriolic Best Buds. Slayer also has a friendly rivalry with President Gabriel.
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.
Masato and Kengo in Little Busters!. Though they're both part of the group of True Companions that are the eponymous Little Busters and often show that they care about each other, both are very competitive, to the point where Kyousuke is able to convince Masato to take his side in the baseball game in Rin's route by appealing to his need to prove himself over Kengo. Sasami and Rin also stumble into one of these, though both the rivalry and the friendship are mostly on Sasami's side.
In Setzer's backstory in Final Fantasy VI, him and Darill were good friends who loved competition with one another. Sadly, she died in a crash long before the story began, setting up Setzer's drinking and gambling issues.
Acrylic and Fetus from Next Breed Of Thief have this. Acrylic, in a somewhat rare bit of humor, refers to Fetus as his "frenemy".
The Sharkasm Crew features one between Cephalo the Pod and Alpha Delpha. It has recently been downplayed as more players of the same skill level as Cephalo and Delpha arrive on the St. Marys scene.
Rainbow Dash apparently forms these kinds of friendships a lot, as evidenced by her interactions with Gilda and Lightning Dust.
The relationship between Pinkie Pie and Cheese Sandwich started out like this, with the latter emphasized on Pinkie's part. That changes when during their Goof-Off, birthday mare Rainbow Dash gets crushed by Pinkie's pinata (thankfully unharmed), making Pinkie realize that her rivalry with Cheese Sandwich is overshadowing her attempt to make Rainbow's party fun and forfeits. Also, Cheese Sandwich was a shy colt who was actually inspired by Pinkie to throw parties. As a token of their friendship, Cheese gives Pinkie one of his rubber chickens that he calls Boneless note which turns out to be one of the Friendship Keys..
This is the purpose of the Equestria Games, which pit athletes from different parts of the kingdom against each other. Rainbow Dash and Spitfire lampshade this with an enthusiastic hoof-bump at the finish line after the relay race.
In the current tennis world, the most famous and friendliest rivalry is between Roger Federer and Rafael Nadal. A very friendly rivalry
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...
One could say that this is the relationship between Nintendo and Sony nowadays.
Said same with Microsoft and Sony, as indicated with this tweet.
The Backstreet Boys and *NSYNC were the two biggest boy bands of the late 1990s and early 2000s with a vicious Fandom Rivalry, but the two groups themselves are close friends and getalong quite well. JC Chasez even produced a song for Backstreet Boys' album Unbreakable and Nick has joked that the two groups should do a concert together.
Historical records indicate that William Shakespeare and Ben Jonson weren't above snarking at each other's work, but each still respected the other.
Pablo Picasso and Henri Matisse had generational differences (Matisse was a dozen years older) and were rivals by virtue of trying to sell to the same collectors. Each one also considered the other about the only artist who could understand them.