My intuition is that it's impossible to guarantee there are no leaks in the LLM as it stands today. It would surely require some new computer science to ensure that no part of any output that could ever possibly be developed isn't sensitive data from any of the input.

It's one thing if the input is the published internet (even if covered by copyright), it's entirely another to be using private training data from corporate water coolers, where bots and other services routinely send updates and query sensitive internal services.