How does a p2p search engine improve the results upon a non-p2p search engine, e.g. Google?

We can imagine different search results for a same request depending on the peers you connect to (if you connect to people you know). You would share a dedicated subset of a search engine with your fellow Cthulhu programming language fans, one with your coworkers, another one with your family, with your bowling club, etc.

Google always searches the same, entire web whoever you are; maybe it is not the best solution?

Yes, whether it is P2P or not is not very relevant to this problem. On the other hand, P2P allows a not-for-profit, ad-free, "do no evil", privacy and freedom of speech friendly search engine (because the infrastructure costs would be shared between the users). I guess a big problem would be the latency.

Exactly, my thoughts would have been to avoid the costs of the data center, and to provide more privacy. Although it is not trivial to protect the privacy in a p2p scenario either.