Naruto's relationships would be significantly altered. With his peers, they might look at him with a mix of envy and admiration. His new Saiyan powers would set him apart even more. His bond with Kakashi could be tested as Kakashi tries to figure out how to train Naruto in this new power set. His relationship with Hinata might also change. Hinata might be more shy around him, seeing him as an even more powerful and intimidating figure. And Naruto's relationship with the other villages would also be different. They might view him as a greater threat or a more valuable ally, depending on how he uses his Saiyan powers.
If Naruto becomes a Saiyan in fanfiction, his relationship with his mentor, like Jiraiya, could be different. Jiraiya would have to guide Naruto in handling these new Saiyan powers. Naruto's friends might be both in awe and a little scared. For example, Sakura might be worried about Naruto losing control of his new powers. And his relationship with the villagers could change too. Some might see him as an even greater protector, while others might fear his new, more alien - like powers.