Unless you are also installing your own personal web crawler that scans all of the web pages of the world and indexes them on your local machine, then something is reaching out to someone else's server somewhere to search their database for the inquiry you are making.
Because your machine reached out to someone else's server, then that someone else knows what you searched for, and can build a database of inquiries from you (or your local install of Searx.)
This is true.
...and can build a database of inquiries from you (or your local install of Searx.)
Except that your local install of Searx creates a brand new, unique, randomized profile every time you perform a search--not just for each browser session.
That makes tracking rather difficult. For this reason, an extra layer of security would be to put this installation in the cloud (like Linode) to hide your own IP.