Developing on devices
Apple's developer kit Xcode or the Android SDK is not necessary to download 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 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 trial version of Corona Simulator allows building Adhoc (for iOS) and Debug builds (Android) for testing on your own devices. The builds will contain a Trial User message box when the app runs on the device. You will need to purchase a Corona Subscription if you want to build for Apple's App Store or Android's Google Play Store. Corona Subscribers also get the benefit of subscriber-only features such as access to Daily Builds, Subscribers Only areas on the Corona forums and LaunchPad.