Rev José Mario O Mandía
This sacrament is unique in that the minister is not a deacon, or a priest, or a bishop, or even the Pope! The priest or deacon who officiates is only a witness. In Matrimony, man and woman are both the subjects (the ones who receive) and the ministers (the ones who carry out or execute) of the sacrament.
The Youcat (261, 1st edition) says: “The man and woman mutually administer the sacrament of Matrimony. The priest or the deacon calls down God’s blessing on the couple and, furthermore, witnesses that the marriage comes about under the right circumstances and that the promise is comprehensive and is made publicly.”
How about in the Eastern Churches? The CCC (no 1623) says: “In the traditions of the Eastern Churches, the priests (bishops or presbyters) are witnesses to the mutual consent given by the spouses (Code of Canon of the Oriental Churches, can. 817), but for the validity of the sacrament their blessing is also necessary (Code of Canon of the Oriental Churches, can. 828).”
Point 1630 of the CCC explains further why a priest or deacon has to be present. “The priest (or deacon) who assists at the celebration of a marriage receives the consent of the spouses in the name of the Church and gives the blessing of the Church. The presence of the Church’s minister (and also of the witnesses) visibly expresses the fact that marriage is an ecclesial reality.”
Furthermore, since marriage has been raised to the dignity of a sacrament, a baptized person can only be married validly if the marriage is done in a sacramental way. The Code of Canon Law (canon 1055, Section 2) explicitly says that “a valid matrimonial contract cannot exist between the baptized without it being by that fact a sacrament.”
The CCC (1631) adds:
“This is the reason why the Church normally requires that the faithful contract marriage according to the ecclesiastical form. Several reasons converge to explain this requirement (cf Council of Trent: DS 1813-1816; Code of Canon Law can. 1108):
“- Sacramental marriage is a liturgical act. It is therefore appropriate that it should be celebrated in the public liturgy of the Church;
“- Marriage introduces one into an ecclesial order, and creates rights and duties in the Church between the spouses and towards their children;
“- Since marriage is a state of life in the Church, certainty about it is necessary (hence the obligation to have witnesses);
“- The public character of the consent protects the ‘I do’ once given and helps the spouses remain faithful to it.”
What about if one of the spouses is not a Catholic? Point 345 of the CCCC (cf CCC 1633-1637 for more detail) teaches:
“ A mixed marriage (between a Catholic and a baptized non-Catholic) needs for liceity [lawfulness] the permission of ecclesiastical authority.
“ In a case of disparity of cult (between a Catholic and a non-baptised person) a dispensation is required for validity.“In both cases, it is essential that  the spouses do not exclude the acceptance of the essential ends and properties of marriage. It is also necessary for  the Catholic party to accept the obligation, of which the non-Catholic party has been advised, to persevere in the faith and to assure the baptism and Catholic education of their children.”