That wouldn't be necessary. The indexing service could reside on your computer with the "omnibar" able to access it to find results (either built into Chrome or via extensions to other browsers).
But since Google would probably want to know the content of your documents to better target ads, the omnibar would eventually, probably, become the nightmare you speak of.
It would be necessary, why search for files on a second computer if you don't want access to those files? Unless the search is restricted to one folder then the whole hard drive is open to Google. I remember Google Desktop being similar to what he wants other than secondary computer access.
But since Google would probably want to know the content of your documents to better target ads, the omnibar would eventually, probably, become the nightmare you speak of.