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.
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.
One possibility is that there was some sort of spiritual or chakra - based connection between Naruto and Nagato. Nagato, being the possessor of the Rinnegan, recognized Naruto's strong will and the fact that he was on a path to change the ninja world for the better. Through this connection, Nagato was able to transfer the power of the Rinnegan to Naruto. It could also be related to the nature of the Sage of Six Paths' power that was somehow involved in this transfer, as both Naruto and Nagato were related to the Sage in different ways.
Another possibility is that it's a result of some strange chakra experiment gone wrong (or right). Scientists in the Naruto universe are always dabbling in chakra research. Maybe an experiment meant to enhance her natural abilities instead gives her the Rinnegan, and she then has to learn to control it quickly to avoid being consumed by its power.
Well, for Sasuke's Rinnegan to be involved in time travel in fanfiction, it might be linked to some ancient jutsu or hidden knowledge. For example, there could be a long - forgotten scroll that details how to use the Rinnegan for time - related feats. Sasuke, being the inquisitive and powerful ninja he is, could stumble upon this knowledge. Then, he might have to face the consequences of meddling with time, such as changing the course of events in the Naruto universe.
Their relationship could start as a rivalry. Fem Sasuke seeing Naruto with the Rinnegan as a new challenge to overcome. But as they face challenges together, they start to respect each other. Eventually, they could become friends and allies, relying on each other's strengths in battles.
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.
In some fanfiction concepts, it might be due to a unique chakra interaction. Maybe the Kyuubi's powerful and ancient chakra somehow triggers a latent ability within Naruto that results in the manifestation of the Rinnegan. It could also be part of some kind of prophecy or destiny within the fan - created world of the story.
He could become more confident, bordering on overconfident at times. With the Rinnegan, he has access to incredible powers, and this newfound strength might make him believe that he can handle any situation alone. However, his friends would surely keep him in check. For example, he might initially rush into a battle thinking his Rinnegan - enhanced abilities will ensure victory without fully considering the risks, but then realize his mistake when his friends point it out.