Calibre is used as a server all the time, see calibre-web.
calibre-web is technically not Calibre and is written and maintained by different people, although it does use the Calibre database (and I believe it must be created with desktop Calibre initially). But it's a good option and I highly recommend it.
I'll add that it's a meta-search engine rather than something that does the actual searching itself. That's still useful, but you're limited by the quality of the upstream search engines (including google, duckduckgo, qwant, etc).
The gain from self-hosting is that you have more control over the results, and can do things like redirect social media sites to privacy friendly alternatives, and create your own bangs (or even add your own custom search engines).
Probably not a massive privacy gain though, although if you host the instance behind a privacy VPN queries won't be associated with your IP at least.