I followed the manual:
Getting started with .NET Mobile Services authentication for the Windows Store
I can start the service locally until I need to authenticate the user. I can also authenticate the user if I publish the Azure service. But I want to verify and authenticate the user locally. How can I do that?
I am using a Live ID and I have the correct ClientID and ClientSecret installed in Web.config. When I try to call LoginAsync on the client, the call fails. The request could not be completed. (Method not allowed)
** Update 2014-03-20 **
Based on the comments of Carlos and Henrik, I updated my local service to look exactly like my server instance. I followed Scot Hanselmans with an excellent guide , and now I have my service running locally on port 80 and port 443 with a fully valid SSL certificate. It even runs on the same https://xxxx.azure-mabile.net hostname.
With these changes, there is now no difference in configuration between running the application on my local computer or running it against Azure. I can go to https://xxxx.azure-mabile.net in a browser, get a redirect to Live login, log in and redirect back to the service successfully. Everything works in the browser. However, it still does not work in the application.
I hooked up the debugger, set the CLR errors for "break when throw", and I managed to catch the exception in the service. Here is what I see in the next window:

The Response property is not useful. It does not contain additional information about the problem.
The only thing that stands out for me is that the application tries to POST before / login / microsoftaccount , while the browser usually does GET at this address (then redirect).
** Update # 2 2014-03-20 **
Following the Henriks manual for remote debugging , I was able to load characters and get a bit more information:
"An existing connection was forcibly closed by the remote host"
Error code: 10054 (WSAECONNRESET) Connection reset using a peer-to-peer network.
It seems the Live Authentication server can force the connection to end, but only when I authenticate the application. Again, browser authentication is wonderful. This, combined with the fact that / login / microsoftaccount is POST from the application, seems to indicate that there is a problem with the authentication token from which I am returning from LiveClient.LoginAsync. I will do even more digging ...