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I know for a fact that the wake word processing is done locally on the device. You could even check its network traffic to see that this is the case.

I'm not sure what protections you would have against a secret court order making the device always listen (via an OTA update) for select individuals, but you can at least check that they don't have such monitoring enabled for most devices.

> if and only if, the "wake word" processing is done locally

That's what it's doing.

Wake-up word is the way to go for many reasons.