Most importantly, this person is a 'communication link' between the pushback tug operator (hand signals) and the cockpit (headset) and also the outside set of eyes for the cockpit crew. Naturally, any abnormalities during engine start must be reported to the crew.
In standard pushback procedures, normal verbiage will form cues to release and set the parking brakes and to confirm when it is clear to start the engines (mind the jet blast risk to other ramps on tight taxiway turns). In the company I work for, it is also common for this person to disconnect the towbar from the nosewheel and tug (in case of a towbar tug; towbarless is much safer though).
In some cases the tower may instruct the cockpit crew to deviate from standard pushback procedures, for instance to do a long push-pull, or to push the aeroplane abeam a specific ramp. This must of course be relayed to the pushback tug operator.