It could completely change the power dynamic in the story. Naruto with the Rinnegan would likely become almost invincible, making his battles much less challenging.
If Azula had the Rinnegan in fanfiction, it would change her character arc significantly. She might start to view power in a different way. With the Rinnegan, she could potentially gain access to knowledge about different worlds, like the Naruto world. This new knowledge could either make her more power - hungry or cause her to have a change of heart and use her powers for more noble purposes. She could also use the Rinnegan's ability to revive the dead to bring back someone important in her life, like her mother perhaps, which would add a whole new layer of complexity to her story.
They would likely be extremely powerful. With the Rinnegan, they could potentially have abilities like seeing chakra in the Harry Potter world, which could be translated to seeing the flow of magic in a unique way. This would give them an edge in duels and in understanding magical spells.
In a fanfic, Naruto with the Rinnegan and Sharingan would have a whole new set of challenges. He would have to deal with the jealousy and fear of other ninja villages. His own village might also be divided on how to handle his new powers. On the positive side, he could use these eyes to better understand the history of the ninja world. For example, the Rinnegan might give him insights into the Sage of Six Paths' true intentions, and the Sharingan could help him uncover secrets about his own family. He would also have to train hard to fully utilize the combined potential of these two powerful eyes, which could lead to some interesting training arcs in the fanfic.
In a fanfiction scenario where Naruto has Rikudou Sennin's powers, Kyuubi's chakra, and the Rinnegan, his power level would skyrocket. The Rikudou Sennin powers would enhance his understanding of chakra and life itself. The Kyuubi's chakra would boost his physical and chakra - based abilities even more. And the Rinnegan? That would open up a world of new jutsu possibilities. He could potentially reshape the Naruto world as we know it, perhaps even finding new ways to bring peace between the villages.
One implication is that he becomes much more powerful. The Rinnegan gives him access to powerful abilities like the Six Paths techniques, which would change the power dynamics in battles in the fanfiction. He could now face stronger enemies more easily.
If Naruto had the Sharingan and Rinnegan in fanfiction, his combat abilities would be off the charts. With the Sharingan, he could copy his opponents' jutsu much more effectively than before. And the Rinnegan would give him access to powerful abilities like the Six Paths Techniques. He might be able to control gravity, absorb and release chakra in new ways, and have enhanced perception of his surroundings.
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.
If Naruto got the Rinnegan, he might become more cautious. The power of the Rinnegan is so great that he would realize he needs to be more careful with it. He wouldn't be as brash as before.
It could be that he was in a life - or - death battle against a powerful enemy. Maybe he was pushed to his limits, and in that moment of extreme stress and desperation, his latent power awakened in the form of the Rinnegan.