Well, they could be quite confused at first. Percy and Annabeth would be out of their element in the wizarding world, and vice versa for Harry. But then, their shared traits of bravery and loyalty would bring them together. Maybe they'd exchange stories about their battles against evil. Percy would talk about his quests for the Greek gods, and Harry would share his experiences with Voldemort. And who knows, they might even learn some new skills from each other, like Annabeth learning a bit of magic or Harry getting better at combat without a wand.
In a fanfiction, they might meet in a dangerous, enchanted forest. Percy has been lost in there for days, and Annabeth has come to find him. As Percy hears the rustling of leaves, he gets ready to fight, but then he sees Annabeth emerge from the bushes. Annabeth quickly assesses the situation and starts leading Percy out of the forest, all while they catch up on what has happened since they last saw each other. There are strange creatures lurking around, but they feel safer now that they are together.
Percy would probably be curious. He's always up for new adventures and meeting new people. He might immediately ask Harry about his magic and how it works. Annabeth would be more cautious. She'd be observing Harry, trying to figure out if he could be trusted. But once she realizes he's on the side of good, she'd start asking intelligent questions about the wizarding world, like how their education system compares to Camp Half - Blood's training.
There could be a school janitor who is actually an old demigod who retired from the hero business. He could give Percy some valuable advice based on his own experiences. And of course, Percy will surely meet Annabeth's classmates, some of whom could be friendly and some not so much, depending on how they react to the new boy in school.
Grover could be one of Percy's friends to meet Annabeth. He's a satyr and has been with Percy through a lot. When he meets Annabeth, he might be really happy to see another important person in Percy's life.
One possible plot could be that Percy and Annabeth accidentally join the same online gaming group. They start off as rivals within the game, constantly trying to outdo each other in quests and challenges. But as they communicate more through the chat function, they start to realize they have a lot in common. They gradually become friends and start sharing their real - life problems. Eventually, they plan to meet in person at a gaming convention.
One possibility is that they meet by chance in a big city. Percy might be working as a marine biologist and Annabeth as an architect. They could initially not recognize each other due to the years apart. Then, as they start talking, the old memories and feelings start to resurface.
In fanfiction, when Percy and Annabeth meet, they might have a super sweet reunion. Maybe they'll share a warm hug and exchange some inside jokes that only they understand. It could be in a new and exciting setting, like on a mysterious island they've never been to before.
In many fanfictions, when Annabeth's friends meet Percy, there could be a mix of reactions. Some might be initially cautious, especially if they've heard of Percy's dangerous adventures. But often, they end up being impressed by his skills and his loyalty to Annabeth.
In fanfiction, they might first be a bit shocked to see each other. Then they could start reminiscing about their old adventures. Maybe they'll decide to team up again for some college - related mystery.