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If the web-page's javascript ONLY stores and processes data stored in the client's localStorage to generate the local page, and sends nothing back to the server, so the web-site operator never sees that data, then is the web-site operator processing that data, or is it only the user-agent's operator ?
The web-site operator certainly wouldn't be a "data controller" since it isn't collecting or storing the data. And it's hard to see how the web-site operator would be a "data processor" in that circumstance.