The software lives on each speaker. The controller is merely a remote control that reaches out to that. There is some memory in each Sonos device that lives across power cuts, and stores all the data. 
 

Each speaker is a three in one, a speaker, a computer, and a WiFi card, with all the things associated with that. 

Thank you for your very helpful reply. One more question. If the speaker is unplugged for a lengthy period of time (for example, a month), will it remember the configuration when it’s plugged in again?

Yes. It should.