Reasonable thing to want a straight answer on.
Stays on your machine, always: what you type, what the model answers, the model files, your conversations, and your documents.
Does leave, and only this: signing in exchanges your account identity and which plan you're on. Update checks ask the website what the newest version is. Downloading a model fetches it from Hugging Face, as you'd expect.
Not sent anywhere: your prompts, the model's output, your files, or details about your hardware.
Once a model is downloaded, you can disconnect entirely and it keeps working — a fair way to verify the claim yourself.
Written up here: https://lizard-llm.qendryx.com/faq.html