One always has to be suspicious of Mozilla these days.

As far as I know Mozilla never lied on technical details, at worst they didn't know about wrongdoings by their partners and acted once known about.

The translation being done locally is advertised and should be simple to check with it being open-source. It also has to download for a while on first use, as well as translations take a considerable amount of time depending on the hardware. E.g. for larger sites it takes a short while until the last paragraph shows up as translated.

It uses the Marian library via WASM (their wrapper for this is here) to do translations, which AFAICT is "AI" based (which I presume knocks the file size down quite a bit) - additionally, the language packs (I'm not sure what term to use here so I'll just go with that) are not all bundled with Firefox, they're downloaded when you first use them.

The previous incarnation of it, the Firefox extension's repo was found over here - I assume the code is now within Firefox's main repo since its built into Firefox now.

I think the way to know is to unplug the PC from internet and see if Firefox can translate stuff.

Only their word until someone does it with a sniffer. E: I suppose, or looks the source but someone answered better now.

Note: Unlike other browsers that rely on cloud services, Firefox keeps your data safe on your device. There's no privacy risk of sending text to third parties for analysis because translation happens on your device, not externally.

https://support.mozilla.org/en-US/kb/website-translation