So I can confirm that the specific Google Nest Thermostat referenced in this post (the 2020 learning model with the mirror display and capacitive touch bar) can be connected to Home Assistant via the Matter protocol, but cannot be operated 100% locally without an internet connection. I suspect the same for the rest of the Google/Nest thermostats.
The thermostat checks for an internet connection during it's boot process, presumably for provisioning and to apply initial settings. If it can't dial out, all smart functionality (including matter protocol) is disabled and it reverts to acting as a 'dumb' thermostat. Even if you let it boot up with an internet connection, it performs period checks/syncs so once that internet connection is blocked (via firewall or other means) it's just a question of when, not if, the device's matter connection will stop working.
Best guess is that to keep the production cost low and profit margin high, the onboard hardware/software is not capable of handling all the logic of schedules, smart functionality, etc locally so it has enough power to send/receive data from Google's cloud services and update the controls accordingly but not much beyond that and likely all of the matter functionality is being handled via the cloud services as well then simply relayed to/from Home Assistant via the matter protocol. Which is a massive bummer because it means that we're still at the mercy of Google. When (not if) they decide these devices are no longer supported, unless someone gets crafty and creates a custom firmware, or Google suddenly gains a conscience and releases an EOL firmware update exposing basic 100% local matter functionality, these devices become dumb thermostats with no integration possible at best or (probably more likely) ewaste.
It's especially disheartening when the goals of the matter protocol were universal compatibility and fully local control, only for manufacturers to use sh*tty implementations like this. I was incredibly excited when I heard that my thermostats were getting matter support, only to be let down by Google's implementation. I love the look of the Nest thermostats, but when these become EOL I will definitely be looking at ugly bricks if it means I can get 100% local control.