You don't need to use an IDE in order to write Xamarin.iOS or Android apps. You can just use a text editor (although Xamarin Studio is free).
The MSBuild logic, at least for Xamarin.Mac and Xamarin.iOS has been made completely open source, so all the logic to build your apps + the SDKs are all open source.
Microsoft has not open sourced Visual studio (VS) so similarly they will not open source Xamarin. You can also use .net framework without using VS. Just use command line tools
The MSBuild logic, at least for Xamarin.Mac and Xamarin.iOS has been made completely open source, so all the logic to build your apps + the SDKs are all open source.