Fair catch, I put that too loosely. The queries do leave your box: SearXNG is a metasearch proxy, so it forwards each search to the engines you've enabled (Google, Bing, DuckDuckGo, etc.).

What stays local is everything about you. Per the SearXNG docs, it strips cookies and sends a random browser profile on every request, so the engines see the query plus your server's IP, not your browser, account or search history. The real win of self-hosting is that you control the logs instead of trusting a public instance's admin. https://docs.searxng.org/own-instance.html

If you don't want even your home IP attached, settings.yml has outgoing: proxies (use socks5h:// for Tor) or source_ips. https://docs.searxng.org/admin/settings/settings_outgoing.html

For a query that truly never leaves the box, you'd need a local index, which SearXNG also supports via its offline/local engines.