I do not run Immich but their privacy policy states:
Locally Stored Data: Immich stores all your photos, albums, settings, and other metadata locally on your device. We do not have access to this data, nor do we transmit or store it on any of our servers.
https://immich.app/privacy-policy
Additionally in their docs I can find no mention of connecting to outside services. I do see references to downloading hugging face models.
https://docs.immich.app/features/facial-recognition#how-face-detection-works