You want the capture and text workflows with zero third-party network traffic.
Point Omni at a local model you run yourself and nothing leaves your machine; the same actions work, subject to the model's own quality and speed. Storage follows the same philosophy: on the free plan, templates live in local storage on that machine, and your API key sits in chrome.storage.local, never in sync storage.
Point the text tier at the local model in Settings, Models, stay on the free plan if you do not want an account, and export templates from the templates editor when you want a backup or a move to another machine. Before relying on image actions, check Settings, Overview: image models must be vision-capable, and its live check confirms whether yours can read images.