As you, I needed to build locally my application, principally to debug once Intel XDK, at least in the version 1621, does not provide support to load third-party-plugins(eg: PhoneGap Push Plugin) on Intel App Preview debug mode. Another problem was generate an iPhone build for beta tests.
My solution was, as the others suggested, to create an similar cordova project and copy the main files from my Intel XDK Project, www folder to be more specific. Config files will be found in platforms folder once you build using cordova/phonegap.
It allowed me to build for android on my machine. Debug was easy using "Chrome Inspect" because cordova generates an debug-unaligned.apk.
Allowed me to have access to the iOS build files, this is a good thing to do if you want to build using Xcode instead.
I Hope it help you.