How are they listening to a few minutes before the button? A "temporary" recording of the first 2 minutes of every call? I feel like there would be some privacy concerns.

It'd suffice for them to sign the transmitted information (audio and required timing metadata for the packet stream) and make you do the temporary recording on your side, transmitting it back to the provider once you hit the button, to let them handle the backup/safekeeping aspects for you.