Since Azure is about services your product designer can put it all in the cloud or use a hybrid approach where your app calls the Azure services. So it's all up to you how it works.
I worry that you wanted to build an Azure app but not use Azure servers. That doesn't make sense does it.
I am part of a group of people developing an service on an Azure server. The simple question is, Does a program, or does the code more specifically, live on the Azure server. If so, is there a way to host the code on a personal or business PC? If the code lives on a PC or Business Computer, how would one go about changing that host?