AppBuilder does not provide an option to build projects locally. A local build would require you to download, install and setup many libraries like the Android Developer Tools and the iOS SDK along with Cordova plugins and more. Also, in order to build iOS apps you will need a Mac machine. AppBuilder simplifies this by setting up a cloud build, so you do not need to make these configurations on your machine and lets you build apps for various platforms regardless of the OS you are using.
If you are concerned about your project being stored in the cloud, then you should use the AppBuilder VS Extension or the AppBuilder CLI as they use the cloud just for the build and do not store your project afterwards.
Let me know if this works for you.