No it doesn’t. “Standard remote identification” drones would be required to have a base station that’s capable of connecting to the internet, but the drone could still operate where there’s no service:

> If the internet is available at takeoff, the UAS would have to do the following from takeoff to landing: (1) connect to the internet and transmit the required remote identification message elements through that internet connection to a Remote ID USS; and (2) broadcast the message elements directly from the unmanned aircraft.

> If the internet is unavailable at takeoff, or if during the flight, the unmanned aircraft can no longer transmit through an internet connection to a Remote ID USS, the UAS would have to broadcast the message elements directly from the unmanned aircraft from takeoff to landing.

However, for “limited remote identification” drones, which lack the hardware to broadcast directly and thus broadcast only via the internet, the base station would be required to have an internet connection [edit: and also be within 400 ft of the drone].