In reference to the Eucharist as a sacrifice, the communion, under both kinds, of the celebrating priest belongs at least to the integrity, and, according to some theologians, to the essence, of the sacrificial rite, and may not therefore be omitted without violating the sacrificial precept of Christ: “Do this for a commemoration of me” (Luke 22:19). This is taught implicitly by the Council of Trent (Sess. XXI, c. i; XXII, c. i).
As our Lord is whole and entire in both the chalice and the hosts after the consecration, a priest who is not celebrating the Mass, or a layperson is not obliged to receive both “species” as theologians refer to the consecrated wine and bread. In His glorified body it is impossible for Jesus to be diminished, so we do not have to worry about “missing out” if we only receive one or the other elements in the Blessed Sacrament.
So the majority opinion of theologians and the legislation of the Church is as the priest offers the sacrifice in “Persona Christi” or in the person of Christ, he must complete the Sacrifice both in reality and symbolically by reception of the host and the chalice. I know some priest who offer the Latin Mass who use a very small amount of wine, just what is needed (10-15 ml), as they do not have a liking for wine.
Some of the reasons why in the Western Church communion of the Faithful and of a priest not celebrating the Mass was limited to only the host included the fear of profaning the Sacrament through spilling the chalice, hygiene, and most important the assertion of early Protestants that everyone Must receive both species in order to receive the Sacrament. From the earliest days of the Church when communion was taken to the sick it was only the consecrated bread, and not the chalice that was taken, if the chalice was needed to make a complete Christ for the Faithful, it surely would have been required for the sick.
Even in the Eastern Rites, communion of the sick is only the consecrated bread. Because of the manner they distribute communion, the priest does take a small chalice, spoon and some unconsecrated wine to give communion to the sick, as the Byzantine custom is to use Zyme, or yeast bread and it would be more difficult for a sick person to swallow the Eucharist if it was dry in the form of a small cube of bread.