The app will ask for location preferences but only uses this to determine local sunrise and sunset times. This does not need to be enabled to use the app. If location services are enabled for Day & Night it will never be sent off device.
The only data that is saved to disk is presented in the preferences pane and the main screen. If you want to verify that check in
~/Library/Containers/org.brandonroehl.DayAndNight/Data/Library/Preferences for the preferences and ~/Library/Containers/org.brandonroehl.DayAndNight/Data/Library/Application Support/Day & Night for the images. No data is sent off besides what is collected from Apple if you decide to share statistics with developers.
The app only connects to the internet though the App Store when verifying or purchasing premium content.