I'd imagine as soon as the message leaves the "bubble" of space that contains the spaceship, it'd continue travelling at normal light speed. So while yes for the split second that it takes to travel from the ship to the drive's event horizon it'd be travelling at FTL speeds (relative to the outside observer) it won't arrive anywhere ahead of the ship. Indeed, anyone receiving the message would probably experience some weird doppler shift effect. Indeed, if I'm figuring this right, if the receiver's position is in front of the ship, when the message does eventually get to them (long after the ship itself) it'd probably be completely back-to-front and the ship would be constantly over-taking the message as it's transmitted. It'd be sort of like firing a water pistol out of a super sonic jet...and about as useful.
So in short, radio communications with an object travelling at FTL is just downright impractical. But then seeing where you're going at those speeds would be just as difficult since by the time you see an object, you're already on top of it. Not so much a problem with the big slow moving kind like stars and galaxies, but asteroids, comets and dust belts could be a serious hazard.
AFAIK the best theoretical idea for practical two-way FTL communications is in manipulating twinned sets of quantum entangled particles. On the one hand the range is infinite (at least within this universe) and the only lag would be due to bandwidth limitations (since you can only "transmit" one binary bit of information per particle) but you'd only be able to communication through those two twinned nodes. So for every ship sent out, you'd need one dedicated communication node back on Earth and the only way for ships to talk to each other is to either have an extra node set per ship (which gets exponentially more complex the more ships in the fleet) or you're dependent on Earth to relay any messages back and forth, which will tie up the nodes at both ends.