But we already have safeguards for this. If compute has access to your private data, it shouldn't have unfettered access to the Internet. Sandboxes like Claude's have had rules like this baked in forever (they only whitelist package repos AFAIK).
From a UI perspective this is also easy to design around. You need an open Internet task, you prevent the task from reading existing files. The user tries to upload a file to such a task, and a prominent warning explains the rare but possible risk.