No, it doesn’t. You may have been understandably confused by the example given in the supportbase article about local processing, which gives the LIFX bulbs as an example of something which doesn’t run locally.
But while there are now a significant number of stock Device type handlers which can run locally (and the same ones can run locally on the V2 hub, btw), No custom smartapps run locally, no custom DTHs run locally, webcore does not run locally, modes do not change locally, routines do not run locally, smart home monitor cannot be armed or disarmed locally except for the ADT model, and the phone app cannot communicate locally with the hub.
The only code which does run locally is the official smart lights feature (and then only if all of the devices are eligible to run locally) and some parts of the official smart home monitor.
Some preconfigured automations can run locally.
Manual, on-demand control of a device or SmartApp through the SmartThings mobile app always requires an internet connection to the cloud and cannot be performed locally.
All notifications, including those from smart home manager, cannot run locally, again with the exception of some features of the ADT system.
Smartthings remains primarily a cloud-based system.
You can verify what runs locally on your own account by checking the IDE.