The company behind a chatbot can store what you type. Depending on the service and your settings, the company may also use it to train AI. Cross says the risks go beyond that.

Cross pointed to recent cases in which information shared with a chatbot appeared elsewhere online. "Another risk is that it's possible information you give a chatbot could be surfaced in other people's google searches," Cross said. That has happened with some shared chatbot conversations. It does not mean private chats automatically become public.

Sign-up screens leave something out. Cross said chatbots sometimes tell users they will use the data for training, but not that it will also sit on company servers. There, it could show up in other people's chats or become exposed to a hack.

Cross's advice is blunt. "You don't want to give it the highly sensitive stuff that you wouldn't want ending up in the hands of someone else because using those chatbots come with those risks," Cross said. Cross also recommends reviewing a chatbot's privacy settings and limiting the personal details you share about friends and family.