You'd need to customize the Chat ControlTemplate and remove the elements you don't need. I've attached a quick example with the default ControlTemplate ( the part with the entry and the Send button is commented), so you can modify it according to the requirements you have.
I hope I was of help. Let me know if you have any additional questions on this.
I just double-checked the updated sample app and the Chat definition is missing the ItemsSource - after setting the ItemsSource the messages are properly displayed on my side. In addition, you can bind the Author property of the Chat, so that the outgoing messages are aligned on the right.
I've attached the updated app, please download it and give it a try. Let me know if it's ok now.