Why won't peer-to-peer AirPlay work with my iPad / Apple TV?

Peer-to-peer AirPlay allows your friends or family to use their Mac or iOS device to connect to your Apple TV without having to connect to your Wi-Fi network. This only works on newer iOS devices and the 3rd generation Apple TV. Supported devices include the 4th and 5th generation iPod touch iPhone 5, 5c, 5s, 6, 6 Plus, iPad 4, iPad Air, iPad Air 2 and all three iPad mini models. iOS devices need to be running iOS 7.0 or later and all devices need Wi-Fi and Bluetooth turned on for peer-to-peer AirPlay to work.