Well, one possibility is that Joey could be in awe of the Batfam's reputation and skills. He might try to prove himself to them, perhaps by showing off some of his own unique abilities from the Teen Titans. For example, if Joey has some sort of special power like telekinesis, he could use it in a controlled way to gain their respect.
Another scenario could be that the Batfam initially views Joey as a bit of an unknown quantity. They might be cautious around him. But as they get to know him better, they realize he has a lot to offer. Maybe Joey helps them solve a mystery that they've been struggling with, using his different perspective from the Teen Titans world.
It's also likely that there would be some culture shock. The Teen Titans have their own way of doing things, and the Batfam has their very different modus operandi. Joey might find the Batfam's secrecy and brooding nature a bit strange at first, while the Batfam could find Joey's more open and team - oriented attitude from the Teen Titans refreshing.
Joey might be really impressed. He could be like a kid in a candy store, wanting to see how all the cool Batfam gadgets work. For instance, when he sees Batman's utility belt, he might be constantly asking questions about what each gadget does.
Joey could be an ordinary citizen who just happens to be in the wrong (or right) place at the wrong (or right) time and meets the Batfam. Maybe he's a journalist trying to get a scoop on the Batfam and ends up getting more than he bargained for.
In a 'Teen Titans meets Batman' fanfiction, there could be an epic battle. Maybe the Teen Titans initially mistake Batman for a threat and they engage in a fight. Robin, being Batman's sidekick, might have to step in to clear the misunderstanding. Then, they could team up to face a common enemy like the Joker or Slade. It would be amazing to see how Batman's detective skills and the Teen Titans' unique powers combine to solve a mystery or save the day.
Well, there could be some epic battles. Maybe the Teen Titans' powers and the Bat Family's skills clash in a misunderstanding at first. For example, Raven's magic might be misinterpreted by the Bat Family as a threat. Then they'd realize they're on the same side and team up against a common enemy like a powerful supervillain who's causing chaos in Gotham or Jump City.
In the 'Spiderman meets the batfam fanfiction', it's possible that Spiderman would be intrigued by the Batfam's gadgets and their more brooding approach to crime - fighting. He might try to lighten the mood a bit. The Batfam, on the other hand, could be impressed by Spiderman's acrobatic skills and his quippy sense of humor. There could also be some interesting interactions between Spiderman and Robin, as they are both younger heroes in their respective universes.
Well, there could be epic battles. The Batfam's unique skills and the Justice League's superpowers would make for some intense fight scenes. Maybe Batman and Superman have a bit of a misunderstanding at first, with the Batfam trying to protect Gotham's interests while the League has a broader view.
In a 'justice league meets batfam fanfiction', there might be some interesting character interactions. For example, the Flash could be amazed by Robin's agility. Also, Wonder Woman might teach Batgirl some combat techniques. And there could be a lot of banter between the more light - hearted members of both groups, which would add some humor to the story.
Well, perhaps Joey is a big fan of the Batfam in the fanfiction. So when he meets them, he fangirls/fanboys really hard at first. Then, as the story progresses, he gets to know them on a more personal level and discovers that they have their own quirks and insecurities just like regular people, which changes his perspective of them completely.
Perhaps Damian will bring a new dynamic to the Titans. He may introduce different combat techniques he learned from his training. Also, there could be interesting character interactions, like him and Beast Boy arguing over who has the better animal -related skills.
Well, Naruto could bring his unique ninja skills to the Teen Titans. His Rasengan could be a powerful new attack in their arsenal. Maybe he'd teach them some of his chakra control techniques too.