Jerry -- If the satellite constellation were in a geo-sync orbit there would be little to no doppler, but the Iridium satellites appear to be in a polar orbit at 483 miles altitude with a velocity of 16,689 mph; I can't imagine an Earth based Iridium Sat-Phone that wouldn't have doppler coming and going (Up/Down). If the specific satellite you are talking through is at a particular moment in time straight overhead, a moment before it was either North or South of your location. The moment of no doppler would be when you are on a line perpendicular (E-W) to the path of the satellite (N-S).

The more I think about it, I'd be very surprised if the Iridium system doesn't track every Iridium call location. Considering the size of each antenna footprint, it seems to me that hand-off from one antenna to the next and from one satellite to the next would be enhanced by knowing where each caller was located.