here's how they train on your data:

an inference request comes in

claude fable RESTful API service does the stuff, some backend systems run the prefill and batch decode, and your conversation is cached for 5 minutes in some prefix cache.

the request is also sent to claude paraphraser, which does almost exactly the same thing as the compactor and rewrites your conversation.

then they record the paraphrased conversation and train on that. it keeps the salient parts of the conversation, like whatever internal knowledge you have, and disposes of anything that could have been correlated with the earlier conversation, which is easy to do because verification is a string comparison.