He would likely be more confident. Having the powers of the Human Torch gives him a new level of strength. So he might stand up straighter and be more assertive in his actions.
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.
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.
In many fanfictions, the Rinnegan and chakra chains are often intertwined in terms of power and function. The Rinnegan, being one of the most powerful ocular abilities in the Naruto universe, can potentially bestow special properties on the chakra chains. The chakra chains themselves might be a way to channel the raw power of the Rinnegan. For instance, they could be used to bind opponents while simultaneously draining their chakra through the Rinnegan's absorptive capabilities. This connection adds an interesting dynamic to the characters who possess both the Rinnegan and the ability to use chakra chains, making for some really cool and unique battle scenarios in fanfiction.
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 Sasuke. If he were to turn to the dark side in the fanfiction, his Rinnegan powers could be used to destroy Konoha. He has the potential to be extremely powerful and if he has some sort of vendetta or is influenced by some dark force, he might take his anger out on the village.