I think people have a binary understanding on this. Someone is either reading their messages or not. While the reality is that all the messages are read by a machine (not unlike GMail and Outlook) and anything suspicious gets flagged up so a human can read it. This obscure the concept of "reading" as most laypeople understand it.

Summary: don't type in Claude anything you wouldn't like a human to read.

It tells you exactly what it does with your information if you ask it, including this exact scenario, and has for at least four months now (when I asked).