Immich is a really good self hosted option

Synology Photos has a search function. It's not as good as google's though.

Not really local but selfhosted: Nextcloud + memories

Immich. https://immich.app/

AGPL, self-hosted, high feature parity with GPhotos. https://github.com/immich-app/immich/

SpaceInvader One channel on YT has several guides on installing the server on UnRaid and migrating your photo library from Google Photos. https://youtu.be/krAZcUIP3YE and https://youtu.be/LtNWxxM5Mzg

Louis Rossmann recently brought a lot of attention to it by offering the developers a grant via his FUTO org. https://youtu.be/uyTPqxgqgjU

Immich has excellent face detection, as well as search capabilities built-in that can be assisted with locally or remote run ML. https://immich.app/docs/guides/remote-machine-learning/

I've been running it for a couple of months. It's under active development, but I personally feel that it's stable and at a good enough place to be a full replacement of Google Photos. In fact, I turned off all uploads to Google in their Photos app, and rely on Immich auto-uploading to my server instead. Of course that means having a robust (preferably off-site) backup and disaster recovery plan.

Photoprism is working great for me.

I think Mylio does this, even in its free version. I don't think it's FOSS though.

Both Immich and PhotoPrism are great, personally I prefer Immich because it has official mobile apps. PhotoPrism only has an unofficial Gallery app for Android, but requires you to use a proprietary app called PhotoSync to actually synchronize your photos.