At least some of the big companies take this seriously, because I've worked in consultants and seen it. It would be completely reasonable for OpenAI to be more private in the paid models - make the weights readonly, encrypt conversations, allow permanent deletion.
For the free models, I'd find it OK to train on the input, but GDPR says users should be able to request data deletion - not sure how viable that is for LLMs, where training frons scratch regularly (without the requested deletions) is probably expensive as fuck.
The internet has been a wild west with lots of privacy abuse, it's good to see some rules and enforcement!