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.
Recording into the recent past is quite common. You implement it by storing the data *to memory only* until the trigger event happens, then you write out what's currently in memory.
It's a common feature in better dashcams--you get say the 30 seconds before the thump that triggered it. Likewise, many high speed cameras that record some short action. They're actually always running, but dumping the end of the data, they only "record" when the trigger happens.