There could be internal antagonists within Naruto's own village. Some of the more conservative ninja in the village might not trust Naruto's new - found skills and powers obtained from Zetsu. They could try to undermine Naruto or even attempt to exile him, fearing that he has been corrupted by Zetsu's teachings.
The Akatsuki could still be the main antagonists. Since Naruto is being trained by Shisui, they would see Naruto as an even greater threat. His new skills under Shisui's training could put their plans at risk. For example, if Naruto becomes better at stealth and infiltration, he could disrupt their operations more effectively.
Well, one possibility could be that Zetsu imparts some unique and dark jutsu to Naruto. Maybe Naruto starts to learn stealth and subterfuge skills from Zetsu, which are different from what he learned at the ninja academy. This could lead to Naruto having an edge in certain battles where traditional ninja skills aren't enough.
It's likely that Naruto's chakra control and power would develop in new and unexpected ways. Black Zetsu might introduce him to strange chakra - based abilities that are not part of the normal ninja training. This could give Naruto an edge in combat, but it also might make him a target. Since Black Zetsu has his own motives, Naruto could be dragged into some dangerous and complicated situations that he may not be fully prepared for.
In a fanfic where Naruto is trained by Zetsu, the story could take an interesting turn. Zetsu has knowledge of the Akatsuki and their secrets. Naruto could learn about the organization from the inside out. This knowledge could help him in his quest to bring peace. He might also gain some physical enhancements. Zetsu is quite durable in his own way, and Naruto could inherit some of that sturdiness. This would make him an even more formidable opponent in battles against strong foes like Pain or Madara.
He may develop a more calculating side. Black Zetsu is all about manipulation. Naruto could start thinking several steps ahead in a more devious way. For example, he might start using people's weaknesses against them, which is completely against his original nature. This change in personality would likely cause a rift between him and his friends who know him as the always - honest and straightforward Naruto.
Perhaps some rogue ninja from other villages. They might see Naruto as a symbol of Konoha's power and want to take him down to weaken Konoha. These rogue ninjas could be led by a powerful and vengeful shinobi who has a grudge against Konoha. So when Naruto is alone on a mission or something, they attack, and that's when Minato steps in to save him.
Maybe some new rogue ninja group that has a vendetta against the Uzumaki family. They see Boruto as a way to strike at Naruto, so they try to harm him, which then leads to Naruto having to save him.
It could be some rogue ninja clans that are after Karin for her special abilities. They might see her as a tool to gain power or information. For instance, a clan that specializes in blood - related jutsu might want to use Karin's unique blood - sensing ability for their own evil purposes.
One possible antagonist could be a rogue ninja from another village. They might target Tenten for her unique weapon - handling skills, thinking they can use her to create powerful weapons for themselves. Naruto would then have to face this ninja to save Tenten.
The village elders could be antagonists. They might see Naruto and Kakashi as threats to their traditional way of ruling the village and use false accusations to banish them.