The GIRM specifies two sets of guidelines. For Mass without a Deacon, it says:
"163 After the distribution of communion, the priest himself immediately consumes at the altar any consecrated wine which happens to remain; but if there are extra consecrated hosts left, he either consumes them at the altar or carries them to the place designated for the reservation of the Eucharist.
The priest returns to the altar and collects any remaining particles. Then, standing at the side of the altar or at a side table, he cleanses the paten or vessel over the chalice, then cleanses the chalice, saying inaudibly: Lord, may the food we have received, and dries it with a purificator. If this is done at the altar, the vessels are taken to a side table by a minister. It is also permitted, especially if there are several vessels to be cleansed, to leave them suitably covered on a corporal, either at the altar or at a side table and to cleanse them immediately after Mass following the dismissal of the people. "