Sasuke might be the first to discover it. Since they are rivals and friends, he would be very attuned to Naruto's changes. Sasuke has his Sharingan which can detect power differences, so he could sense the new power of the Rinnegan within Naruto.
It could be Hinata. She is always observing Naruto from afar. One day, when Naruto is using his chakra in a new and different way, she might notice the unique pattern of the Rinnegan and figure out that he has this new power.
Well, it could make Naruto extremely powerful right from the start. With the Rinnegan, he might be able to master all types of jutsu more quickly. He could also have an edge in battles against powerful foes like Madara or Kaguya. Maybe he'd be able to use the six paths techniques with more ease and less training.
In a fanfiction where Naruto secretly has the Rinnegan, he would become an even more formidable force. His combat abilities would skyrocket. He could potentially use the powers of the Rinnegan, like the Six Paths techniques, in addition to his existing Sage Mode and Nine - Tails chakra. This would make him a major threat to any enemy. He might be able to travel between dimensions more easily and have better control over life and death, just like Madara and Sasuke did with their Rinnegan.
His relationship with Sasuke would likely become more complex. Sasuke might be both jealous and intrigued by Naruto's new powers. Naruto would have to prove that he still values their friendship despite his increased strength. With Sakura, she might see Naruto in a new light, perhaps with a bit more admiration as he now has such rare and powerful abilities.
In fanfiction, Naruto might keep the Rinnegan a secret by avoiding using its powers in public. He could also use his shadow clones to create distractions when he feels like the Rinnegan is about to be revealed. Plus, he could tell only a very few trusted people, like Kakashi or Iruka, and ask them to keep it a secret for him.
His relationship with Sasuke would likely be very different. Since Sasuke has also had experiences with the Rinnegan, they might now have a new level of understanding between them. They could train together to master the powers of the Rinnegan or compare notes on how to use its abilities.
Well, it might start with fem Naruto somehow obtaining the Rinnegan. This could lead to her being ostracized at first because it's such a powerful and mysterious dojutsu. But as she learns to control it, she could become a real force to be reckoned with. She might use the Rinnegan to travel to different dimensions and gain knowledge from there to help her village.
Edward is a likely candidate. Since he is so attuned to Bella, he might pick up on some strange behavior or scent that leads him to suspect she is hiding something.
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.
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.