after having a handover
Which is what, I think, the OP wants to know about. We do it like this:
The oncoming controller plugs their headset in the available socket (each position has two sockets). This allows direct speech between the two headsets for clarity and recording of the handover should it be required for training or an incident investigation. The outgoing controller, when convenient, describes what is happening. This will include the air pressure, runway in use, aerodrome status, weather, unusual activity and the present and planned traffic situation. When the oncoming controller accepts the position (maybe after a question or a condition like "turn ABC123 left ten degrees, then I'll take it") the outgoing will unplug and should stay in the room for a minute or two, particularly if it's busy or complicated, in case anything has been missed.
It is worth noting that the oncoming controller is not required to accept the position until they are completely happy with the situation. For example, if I thought two aircraft were going to lose separation shortly after I took over (extreme example, I know) I would make the outgoing controller remedy the situation before I accepted responsibility for the position.