These are great questions, that I would be interested in as well.
I'm guess though, for now, that everything requires Google servers. Google probably keeps a state of all devices in some database on their server. So a starter would be when a state (on/off, etc.) or condition (Ok Google command) is met on an object in the database. Then Google servers evaluate the conditions, and then execute the actions.
Eventually, it would be great if certain scripts could run locally if all necessary devices for Starters, Conditions, and Actions supported Matter without requiring internet. Although, I'm not sure if previous nest devices were built with that in mind from a storage, compute, and energy standpoint. Storing scripts, waiting for starter, executing, etc. You'd also have to delegate which devices store and execute the scripts, etc.
I'm not even sure if the Google Home app works offline yet for Matter devices. Philips Hue does, Nanoleaf using Google Nest border routers does, others do, but Google Home app seems to require internet to do anything.
