This is a great question, in that it made me wonder why the Fediverse hasn't come up with a distributed search engine yet. I can see the general shape of a system, and it'd require some novel solutions to keep it scalable while still allowing reasonably complex queries. The biggest problems with search engines is that they're all scanning the entire internet and generating a huge percent of all internet traffic; they're all creating their own indexes, which is computationally expensive; their indexes are huge, which is space-expensive; and quality query results require a fair amount of computing resources.

A distributed search engine, with something like a DHT for the index, with partitioning and replication, and a moderation system to control bad actors and trojan nodes. DDG and SearX are sort of front ends for a system like this, except that they just hand off the queries to one (or two) of the big monolithic engines.