A group of rogue demigods could be the antagonists. They are jealous of Percy and Jason's fame and power. They form a secret alliance and start causing trouble for the other demigods. Percy and Jason meet while trying to figure out who is behind these attacks. At first, they suspect each other, but then they realize the real culprits are this group of rogue demigods and join forces to stop them.
A great antagonist could be a rogue demigod who has been corrupted by power. This demigod could have discovered a way to combine the darkest parts of Greek and Roman magic and is using it to cause chaos among both camps. He or she might see Jason and Percy as the biggest threats to their plans and target them specifically.
It could be an old enemy from their past, like a resurrected Titan or Giant. These powerful foes might manipulate Percy and Jason into betraying their friends as part of a grand plan to regain power or take revenge on the demigods.
Obviously, the gods themselves would be the main antagonists. Since they are the ones who kill Percy and Jason in this concept. It could be a group of gods, like Zeus, Poseidon, and Hera, who feel that the two heroes are getting too out of hand.
Perhaps a rogue group of mermaids and mermen. They don't like the idea of Triton allying with Percy, a half - blood. They believe that Triton should only focus on the pure - blooded sea creatures. So they become the main antagonists, constantly attacking Percy and trying to turn Triton against him.
Another possible antagonist could be a powerful sorcerer or witch who is envious of Aphrodite's beauty and power. They see Percy as a way to get to Aphrodite. So they try to use Percy as a pawn, casting spells on him to make him do their bidding and cause harm to Aphrodite, all while Percy is trying to figure out what's going on and break free from the control.
In a 'Jason meets Percy' fanfic, they could meet in a place that is neutral to both of them, like a magical marketplace. Percy is there looking for a special item for his next adventure, and Jason is on a mission to find a way to strengthen his camp. They cross paths and initially have a bit of a misunderstanding. But as they start talking, they find out they have a common enemy in the background planning something big. So they decide to join forces and use their unique skills to stop the enemy.
Another possible antagonist could be a rogue Uchiha. One who didn't agree with Shisui's views on peace. This rogue Uchiha could see Naruto as an outsider interfering with Uchiha matters and Shisui as a traitor for teaming up with Naruto. So they set out to disrupt their partnership and cause chaos in the village.
Maybe it's an ancient evil that has been awakened. It targets both Naruto and Konan because of their unique powers. The evil being could be trying to use their powers to break free from a long - held seal. So, Naruto and Konan have to put aside their differences and fight this powerful foe together.
It could be a new monster that is after Percy. Since Percy is a demigod, there are always monsters chasing him. This new monster might be something that Paul has never seen before and it could put both of them in danger.
I think Percy would be more dominant. He has a very strong will and a sense of purpose. His connection to the ocean gives him a unique power that is quite formidable. Also, he has a group of loyal friends who support him. Even if Jason is jealous and tries to assert himself, Percy's natural leadership and his ability to think on his feet would likely make him the more dominant character. For instance, in a battle or a difficult situation, Percy would probably be the one making the decisions and leading the way, while Jason might be more focused on his jealousy and trying to compete with Percy rather than actually taking charge.