If you are not a registered Apple developer, but are just using Xamarin and eventually Xcode without yet thinking about going to the store, you won't have the necessary certificates on the OS X side to actually sign the binaries. You will quickly run into compiler errors telling you that your application can't be signed.
This is fairly simple to fix; all we need to do is manually edit the project file:
With the build host running on the OS X side of things and Visual Studio that is paired with it, as described in the Xamarin documentation, you should now be able to run the project using Ctrl + F5 (Debug | Run without debugger). It should yield the following result...