How? Let's suppose I trust Google to crawl the Web and present a plausibly-accurate result to my query. This works because I communicate with Google using TLS (so I trust that Google is in fact Google), Google is the one performing the crawl, and I put my faith (for purposes of this exercise) in Google's editorial discretion and good judgement.
How does a distributed search work? What does my query look like? Where do I send it? How can I gauge how much I trust the response?
The search index could be build using blockchain technolgy. Each transaction is an entry of the search index. For efficiency, such an index could then be mirrored by many different institutions. For example, universities. Each snapshot of the search index could have something like a sha256 sum. If the server you use per default returns a hash that differs from other institutions you rate trustworthy then you should be skeptical.
As for search queries, since the index should be standardized in some way there could be many open source apps that require their own syntax for queries so that they suit different purposes and preferences.
It would be a technical stretch to expand this to replace Google, but if you did you'd have many competing "search engines" built upon an open decentralized database.
I don't think I've expressed this idea well, but you should get the gist.
Sure, but why should we rely on a single middleman who at some point could abuse his power? Something like Google search could possibly work in a decentralized fashion.