A lot of the information out there suggest that ground stations are not required.
This system uses cross-link technology, a unique technology that provides redundancy by allowing satellites to talk to each other. Therefore, when placing an Iridium-to-Iridium call, a user never depends on landlines - ideal for emergency management situations when infrastructures are not reliable. With optimal conditions (line of sight), Iridium has 99.8% call completion.link
These downlinks are primarily used for calls that need to be routed into the terrestrial phone system. For satellite phone to satellite phone calls, it is possible to route the traffic directly through space with no downlink. Although the satellite to satellite routing and the associated downlinks add a considerable degree of complexity to the Iridium system, it does enable fewer earth stations to be required, and it also enables full global coverage to be provided.link
The Iridium network call process is different than the Globalstar network. Unlike Globalstar where all the switches and most of the call handling is made from the ground station... most of the switching and call handling is made by the Iridium satellites themselves. Iridium to Iridium calls are handled in this way, calls will go up to the Iridium satellite then may travel from Iridium satellite to Iridium satellite till it reaches the receiving Iridium satellite phone never using the Iridium ground station or the public phone network. With this type of call the Iridium constellation completely handles the call. link