I get worried with all this new apps and privacy

what are your specific concerns?

All these apps require screenshots and screen access. I wonder if OpenAI or Anthropic got hacked, and there was a data breach where all this would go. A scenario is similar to 23andMe.

Just depends on how or if they're storing the data, and if they are, how they're storing it (e.g. de-identified, encrypted). Usually these apps use screenshots to get the context for a prompt and then discard the screenshot. But they still have to get permission and list it as data that they use. It's also the case that if you give some third party app direct access to your email, that's likely much more dangerous, since password resets can be done just by knowing your email. There's absolutely considerable risk with the privacy requirements of these apps, but 1) you don't have to use them and 2) they really won't work that well without access (unless one holds up Siri as the gold standard at trading off privacy and performance).