The obvious ones that spring to mind are Mozilla, DuckDuckGo and ProtonMail, but does HN have any other examples?
https://www.mojeek.com/ https://blog.mojeek.com/2020/12/frequently-asked-questions-a...
Other search engines mentioned here, and many more, are primarily syndication partners of Bing (mostly), Google or Yandex.
https://www.searchenginemap.com/
https://element.io/
Threema and Signal for messengers.
They seem different to other search engines in that they actually have their own index rather than just proxying to another engine (e.g. Bing)