For Azure ML, some people use it fully in the cloud, others do local work in their favorite IDE, usually RStudio, then upload scripts, do their training, opertionalization, etc. on Azure. If you are using standard R, Python (eg scikit-learn) you can do work locally or on AzureML, but if you want to use Azure ML's own algos, then they're available only on the cloud.

There are also some sample experiments + data on Azure ML that you can check out quickly (no log in):

https://gallery.cortanaanalytics.com/experiments