In that example, the mnemonist should be demonstrating the security issue to the government, and not to their friend. We have social taboos for this reason. As an extreme example, I wouldn't greet a person by their penis size after noticing it in the locker room - some information should still be considered private, regardless of how we came to obtain it.
Same with an LLM, when it got sensitive information in its weights, regardless of how it obtained it, I think we should apply pressure/shame/deletion/censorship (whatever you call it) to stop it from using that information in any future interactions.