What is an air traffic controller's role in handoffs?
In most modern ATC units, almost all transfers are conducted as “silent transfers”. A silent transfer is a transfer where all necessary information about a flight is sent in advance to the next ATC sector by an automated system. In this case, the controller does not have to manually talk to the next sector, but will simply instruct the pilot to switch to the next frequency prior to the area of responsibility border.
Where manual coordination is required, it is most likely done by the planning controller (PLC). As opposed to the executive controller (EC), who talks to the planes, the PLC (who will be seated right next to the EC) handles coordination with adjacent sectors via dedicated telephone lines.

There are two main parts of a transfer: the transfer of communication and the transfer of control. The transfer of communication is the actual frequency switch – the instruction for the pilot to switch from one frequency to another. The transfer of control is the transfer of responsibility of the flight from one controller to the next.
Before answering your question directly, let us establish the premises of the question.
Note: this answer refers primarily to European procedures, and is largely based on this EUROCONTROL document
In respect to your question, the transfer of control is the interesting part.
Firstly, we must realise that there are two fundamentally different premises for a transfer of control. Either the transfer takes place from one position/sector to another position/sector within the same ATC unit (for example between two area control sectors in the same area control centre), or the transfer takes place from one ATC unit to another (for example when a flight crosses a state border).
In the first case, the adjacent sector will typically already have access to information about the flight, since both sectors rely directly on the same data source. In the second case, this might not be the case.
Additionally, in the first case, controllers will often be physically seated right next to each other. For units where paper flight progress strips are still used (mainly some approach and tower units), this enables controllers to physically pass the flight progress strip to the next controller.
In both cases, there are four required stages which lead up to a transfer of control (refer to ICAO document 4444, section 10.1):

a) Notification of the flight in order to prepare for coordination, as necessary;
b) Coordination of conditions of transfer of control by the transferring ATC unit;
c) Coordination, if necessary, and acceptance of conditions of transfer of control by the accepting ATC unit; and
d) The transfer of control to the accepting ATC unit or control sector.

Doc 4444 also states that:

ATC units shall forward from unit to unit, as the flight progresses, necessary flight plan and control information.

Now, your question is, to what extend is this data transfer process automated. ICAO document 7030 gives us a clue to the background of the answer:

When so agreed between adjacent ATC units, a computer-assisted coordination process shall be introduced to eliminate the need for verbal coordination of boundary estimates and to reduce the amount of manual data input into ATC computers.

In most modern ATC units, almost all transfers are conducted as “silent transfers”. A silent transfer is a transfer where all necessary information about a flight is sent in advance to the next ATC sector by an automated system. In this case, the controller does not have to manually talk to the next sector, but will simply instruct the pilot to switch to the next frequency prior to the area of responsibility border.

Silent transfer of control procedures require that traffic is predictable and sufficient system support is provided to ATCOs to ensure:
-They are in possession of all elements related to the aircraft profile and
-Should the agreed conditions of the transfer change, they have the means to speak to each other such that new conditions are agreed expeditiously and safely.

Silent transfers are based on the exchange of flight data, which happens through connected Flight Dara Processing Systems, which make use of the OLDI protocol.

The operational use of connections between Flight Data Processing Systems (FDPSs) at ACCs for the purpose of replacing verbal &quot;estimates&quot; is referred to as On-Line Data Interchange (OLDI). The implementation of basic OLDI messages is mandatory in accordance with Regulation (EC) No 1032/2006. These messages are defined by the EUROCONTROL Specifications

What this regulation means in practise is that, within Europe, silent transfers are used more often than not. However, you will also see units where a manual coordination prior to a transfer is required, for example flights crossing the border from certain European countries to certain non-European countries. Where manual coordination is required, it is most likely done by the planning controller (PLC). As opposed to the executive controller (EC), who talks to the planes, the PLC (who will be seated right next to the EC) handles coordination with adjacent sectors via dedicated telephone lines.