We already have the means to execute arbitrary code (JS) or specific database queries (SQL) on remote hosts. It's not inconceivable, to me, that my device "knowing me" could consist of building up a local database of the types of things that I want to see, and when I ask it to do a new search, it can assemble a small program which it sends to a distributed system (which hosts the actual index), runs a sophisticated and customized query program there, securely and anonymously (I hope), and then sends back the results.
Google's index isn't architected to be used that way, but I would love it if someone did build such a system.