Repeated throughout the Dune novels a reader will find "communications room".
Example from Heretics of Dune:
A tall Reverend Mother came from the adjoining communications room. "Mother Commander," she said, "the messages have been sent to <PERSON>, <PERSON>, and <PERSON>."
And about 30 lines later a reader can find this:
The Reverend Mother from the communications room returned, glanced once at <PERSON> and spoke to <PERSON>. "Chapter House acknowledges receipt of your report. They approve your response
Which makes me think they had those working as real time communications in place.
Maybe they were using something like ansible for this?
But then I was just reading Dune Encyclopedia and found this:
The centrality crucial to Imperial administration administration was impossible to maintain when there was no quick method of communication from one end of the Empire to the other — and until the discovery of Holtzman Waves, the only way to send a message was to send a ship, which made for impossibly expensive administrative costs, and diffused the power of the emperor irreparably.
In 7562 B.G., <PERSON> discovered the Holtzman Wave; in 7556 E.G. 45
he passed on information necessary for its use to six planetary systems
Also in 7556, his research was interrupted and he was unable to resume it until his orbit brought him back within panel range of his sun. In his second return (5694 E.G.), <PERSON> kept silent; however, he evidently used this period to discover the curious "existence" of the pointsource node. On his third pass (3832 B.G.), <PERSON> was able to communicate freely after some initial difficulties were overcome, and on that pass he published the theory that led to the development of the twodimensional Holtzman Effect, the famed Defensive Shield.
Theoretically, the range of a Holtzman Wave is infinite; in practice, its effective range is limited by the ability of the sender to aim the wave so that it will strike a large (planetary or larger) mass at or near the receiver. This requirement limits the useful range of Holtzman Wave communications to approximately 250 light-years without re-broadcasting.
(...)
Improvement came with the Holtzman Wave, nationally a mathematical phenomenon causing microcatastrophic folds in real spacetime along a selectable vector, when the vector impacts with matter denser than interstellar hydrogen, it excites that matter to emit long radio waves, which can then be received on normal radio antennae. With interstellar communication now possible, navigation could be enormously improved.
This makes me believe that they were using HE with Communication Rooms to the extent that HE allowed and then the message would reach the target planet via another planet. 489