Does the phone running PhoneLeash need to be turned on, and have cellular service? [Yes]
PhoneLeash is installed on the phone whose texts you want forwarded. This phone needs to be powered on, and capable of receiving incoming texts.
If you are forwarding to an email address, the phone will also need an internet connection (cellular or WiFi).
If you are forwarding to another phone (as text), you don't need an internet connection, except during installing the app.