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.
It could become more complex. As adults, they have different responsibilities and experiences. Their relationship might start off a bit strained as they try to fit into each other's new lives. But eventually, they could find a new level of understanding and respect for each other.
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.
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.
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.
They might go on an adventure together. Maybe they discover a hidden beach or a mysterious cave. They could also have some really deep conversations about their past experiences and their future plans.
In some fanfictions, mortals might meet Percy and Annabeth in a coffee shop. Percy could accidentally use his powers to make a small water fountain out of a cup, and Annabeth would quickly cover it up while the mortals are left wondering what just happened.
They would probably meet their old classmates. People like Grover, who was their friend in high school, would surely be there. Maybe some of their old study group members or teammates from school sports teams.
Well, they might have to hide their demigod powers. For example, Percy could be really tempted to use his water - controlling abilities when they are near a big fountain in front of mortals, but he has to hold back. Annabeth would be using her wisdom to come up with quick lies if the mortals start asking too many questions about their strange behavior or the weapons they carry.
They might start collaborating on a big project. Since they are both intelligent and resourceful, they could combine their skills to solve difficult problems at work.