The sacraments are ordinances of the Church; in other words, they are one of the things that makes church into Church. As such, the authority to oversee their administration, and the teaching that accompanies their administration, lies with the government of the Church.
This is why the Westminster Confession restricts administration of the sacraments to ministers of the Word only, those who have been lawfully ordained to that task. We believe that this is appropriate to good order and government, but it is not absolutely essential to a right use of the ordinance. We hold that if the minister is out of town, it would be far better to have one of the elders administer the Supper rather than have the people go without. And if a hard providence created a difficult situation—some Christian laymen locked in prison for their faith—it would be better for them to contrive a way to observe the Supper than to do without.
At the same time, in ordinary circumstances, the man or men ordained to represent Christ to the people should ordinarily baptize and ordinarily administer the elements of the Supper. This is not because he is a priest, but to keep the liturgical symbolism straight.