Not really. Search engines are a tech so centralized only two of them exist in the west, Google and Bing. There are zero open source search engines of any usable quality. Whereas there are lots of models out there, some free to download.
"only two search engines exist in the west" and "only two search engines in the west are of usable quality to me" are contradictory statements.
The models free to download aren't the models used by OpenAI, Anthropic and Google. You aren't going to get all of OpenAI downloaded to your desktop and running fully on just your hardware.
And in each case (search and AI) the potential to decentralize and maintain "usable quality" is limited by these technologies requiring physical infrastructure at a scale that isn't available to the home consumer.
I mean, they are the models used by Google, at least. Gemma is used by Google and you can download it freely, weights and all. OpenAI has released an open weights model although I don't know if they use it themselves.
They aren't as good as the full fat models but they're plenty useful for many real world tasks. Show me the open source web search engine that I can run locally and that's plenty useful for many real world tasks!