Almost everything is still done in the cloud. The smart lighting automations are eligible to run locally, but only if all of the devices used in the specific automation are also eligible to run locally.
Some of smart home monitor is also eligible to run locally.
So far, nothing else. They said they would like to add more, but they’ve also said that the way the architecture is anything that is eligible to run locally has to be pushed out to all customers in a firmware update. So that has complicated things.
There’s a discussion thread in the developers section of the forum if you’re interested in more details.
There are also links where you can check exactly what you have running which is eligible for local operation: