You only need to be online when you rent the vehicle and unlock it for the first time, be that at home, in your hotel room or a local café as well as when you end the rental. Once that's done, you can enjoy the city in offline mode, until you want to end the rental.
You unlock the rental vehicle with the Donkey app, which connects to the smart lock on the bicycle via Bluetooth.
Please note that when renting an e-bike, you need to establish internet connection throughout your rental. More information on renting an e-bike can be found here.