Local processing is extremely limited at present. Basically the only things that will run are smart lighting with some specific device handlers and a little bit of smart home monitor.
You will not be able to use the mobile app with the hub while it is running locally. (It is technically possible to do that, but that is not how smartthings architecture works). And at present, no custom code of any kind runs, although they have said they would like to add that possibility in the future.
Detailed discussion on local processing