That sounds like it places an undue burden onto the user to determine and explain why data might be personal. Is a particular writing style personal? Something that identifies their IP address, or time zone, or three separate messages that can be used to pinpoint someone's identity or narrow it down significantly?
To build on the Matrix example I mentioned, they give you the ability to "redact" messages but it's your job to hunt them down across their entire platform, and obviously you can't look at any messages in any rooms you've been kicked out of (and I'm pretty sure an API call to redact them, even if you correctly guessed the ID, would be rejected).