the privacy thing is a basic one in that all the search providers know which IP the requests are coming from.

Running a searx instance requires no personal data, from you as the hoster or from users.
It's just a simple docker image for most people (that's what I use for my personal instance)

If you're eager to hide your controversial knitting hot takes, hosting a node for yourself isn't going to protect you from unscrupulous three letter agencies.

Hosting a public node can help other people who are worried about privacy as their request will be lost in all the other requests.
Running a public node is a great thing to do as it helps others.

If you're hosting I would recommend going for the domain owner anonymizer protection if your registrar supports it.