Their relationship would likely become more complex. With the Rinnegan involved, they might have to rely on each other more for understanding its powers. They could become more protective of each other, knowing the power and the danger it brings.
Sasuke's reaction could be complex. At first, he would be enraged. After all, the Rinnegan is a symbol of his power and his journey to gain strength. However, he might also have some mixed feelings. He and Naruto have a deep bond, and he might wonder if Naruto had a good reason for taking it. But overall, his initial reaction would likely be one of fury and a desire to get his power back.
In fanfiction, the Rinnegan is often depicted as more powerful than the Sharingan in many ways. The Rinnegan has abilities like the Six Paths techniques which can be extremely overpowered. For example, it can allow the user to control gravity, summon large creatures, and even revive the dead. While the Sharingan has its own set of cool abilities like predicting movements and casting genjutsu, it seems limited in comparison when it comes to the grand scale of power shown by the Rinnegan in most fanfictions.
Well, it could lead to a major shift in the power balance among the characters. Sasuke would be weakened without his Rinnegan, and Naruto would have to deal with the new power. There might also be some internal conflict within Naruto as he tries to come to terms with using Sasuke's power. Maybe he would feel guilty at first, but then realize it's for the greater good. Also, other characters like Kakashi or Sakura might have different reactions. Kakashi could be worried about Naruto being corrupted by the power, while Sakura might be more concerned about Sasuke's well - being without his most powerful ability.
One idea could be a story where Sasuke, with his Sharingan, discovers a new power related to the Rinnegan that was hidden within Naruto all along. Maybe Naruto has some dormant Sage of Six Paths energy that when combined with Sasuke's powers, they can unlock a new level of strength to fight a powerful, unknown enemy.
In the manga, Sasuke doesn't regain his Rinnegan. The story progresses without this particular change. The power he had before is maintained without reverting or regaining additional abilities related to the Rinnegan.
Naruto would probably win. He has an indomitable will and has always found ways to overcome powerful foes. His experience and the unique powers he might gain with the Rinnegan could give him an edge over Madara.
If it's a crossover with the Fairy Tail guild, Natsu would be an interesting character to pair with Sasuke. Natsu is hot - headed and Sasuke is brooding. Their personalities contrast but they are both extremely powerful. Sasuke's Rinnegan powers could be used in combination with Natsu's fire - based magic in battles against powerful enemies. And they could have some great banter as they try to understand each other's worlds.