There's some info. The Darkspawn DLC, awakened ones like Architect and Corypheus. The horde is drawn to find them by the Calling, the Archdemon's voice in their heads. It controls them as they have no control of themselves. So, without consciousness, they can't be worshipped by the Darkspawn. Only served. But once they have thought, only curiosity or actual interest would draw them as Archie showed no actual desire nor need to find another Archdemon after the last one woke up and started another Blight when he tracked it down.
As for the Archdemons, we have no actual info on if they have thoughts either. Just that they cannot die by any without the taint. But if Cory's anything reliable, they once had some method to speak. And if having The Warden run up to decapitate it didn't even warrant a scream, I don't think they talk after being tainted into Archdemons.