Privacy I: Personal Identifiable Information
As with all things connected to the internet, YOU SHOULD NEVER PUT YOUR PERSONAL INFORMATION ONLINE.
Your privacy is taken bit by bit by websites all the time. Your information is bought and sold online by ticking one user agreement at a time. Every website keeps tabs on you across browsers and across devices. Digital privacy is almost an oxymoron; it doesn't exist.
That being said, if you accidentally enter some of your personal identifiable information in an AI, you shouldn't worry too much. Yes, it is now out there. But, in many ways it is a drop in the bucket considering the literal billions of words that are used to train an AI. It should be impossible (in the most used AIs) to piece your personal information together through prompting an AI.
Privacy II: Prompts and Inputs
Assume that all you enter as a prompt, and the AI's response, will be used to train that AI. It may be possible to work backwards and discover what you wrote, but that would / should be VERY difficult.
So, don't enter anything that you would not want other people to know.