Communication between the control rooms and the studio floor is through production talkback (PTB). Some studios have separate talkback systems for lighting and sound, but many have just the one system. Everyone involved with technical operations needs to be able to hear talkback clearly.
The microphone in front of the director is left open. It is assumed that directors (and production assistants who sit next to them) have so much to say that there is no point in switching it off.
An immediate caveat to new directors: assume that talkback is always live, and someone is always listening. Even if a studio is completely empty with the lights switched off, talkback also goes to other parts of the building (e.g. the transmission ...