It is not necessary to download Apple's developer kit, Xcode, or the Android SDK if you only want to use the Corona simulator. In order to build and test your code on an iOS device (iPhone, iPod Touch, and iPad), you will need to sign up as an Apple developer and create and download the provisioning profiles. If you want to develop on Android, you don't need to download the Android SDK unless you want to use the ADB tool to help with installing builds and viewing debug messages.
The Corona SDK starter version allows you to build Adhoc (for iOS) and debug builds (Android) for testing on your own devices. Corona Pro users also get the benefit of special features, such as access to daily builds, premium features, all plugins, and premium support.