I am thinking of a simple chat application in Window Phone using Microsoft Push Notification Service.
I read a guide at http://msdn.microsoft.com/en-us/library/hh202967(v=VS.92).aspx which talks about creating a client application that receive toast notification.
This is a first time for me to develop a mobile application so I do not get some part of it.
The question that I have is, is it possible for a client (window phone) to send toast notification directly to other client? I have found below image from the web. It seems that I need to have Cloud Application(probably webserver?) to play with notification service.

Would it be possible to build a Window phone application that directly request to MPNS to send notification to other window phone application? (so from the picture above, ‘2’ will directly point to Microsoft Push Notification Service)
If it is not, what is the reason for that?
There is a good explanation of the push notification service here.
From the above link.
In your case you’ll need communication between your devices before any notifications can be sent. This is to know the uri of your phone. A server has to store these uris for all devices registered for notifications. This is why you’ll need an application which sends the xml to the MPNS.