With Ecobee? I don't think so... Unless one is communicating to the Ecobee thermostats via Apple HomeKit, all other communications is via the cloud, as far as I know. I used Apple HomeKit to control my thermostats while our internet was out on Tuesday due to Hurrican Isaias. HomeKit devices work locally, which is pretty nice.
I have two Ecobee 3 Thermostats that I installed to replace two Nest thermostats, one which had failed. I am using Hubitat's built-in Ecobee inetgration, which is definitely cloud based. I do not believe Ecboee has a local API, so I am assuming the community Ecobee Suite is also using the Ecobee cloud (although I would be very happy to be wrong about this 
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