When you use local scheduling
Chrome’s extension storage keeps local task records and settings in that Chrome profile. A task record includes its prompt, the target conversation reference, schedule, retry settings, status, and task history. The optional Cloudflare token is also saved in that profile if you configure Cloud mode.
For Browser tasks, the extension opens or uses the saved Claude chat URL and the content script interacts with that Claude page. When sending proceeds, Claude receives the prompt through your signed-in browser session. The extension does not read or export browser cookies.