I think you're suggesting homomorphic encryption to execute the user's ranking model. Unfortunately, homomorphic encryption is pretty slow, and the types of operations you can do are limited. But it's viable if the data you're operating on is relatively small - e.g. just searching through (encrypted) personal messages or something.
I think you've got the right general idea, but I don't know that it has to be homomorphic encryption. After all, an index of the public web is not really secret, and the user doesn't have a private key for it.
In the simplest case, you could make a search engine in the form of a big, public, regularly-updated database, and let users send in arbitrary queries (run in a sandbox/quota environment).
That's essentially what we've got now, except the query parser is a proprietary black box that changes all the time. I don't see any inherent reason they couldn't expose a lower-level interface, and let browsers build queries. Why can't web browsers be responsible for converting a user's text (or voice) into a search engine query structure?
