In Eastern Catholicism, the priest actually does marry the couple as opposed to the couple presenting the Sacrament to each other.
The absolute minimum east or west per canon law, assuming no impediments, is that the couple celebrate marriage in the presence of two witnesses, with approval of the Church per the canon laws. The extreme case illustrates this.
CCEO Canon 832§1. If one cannot have present or have access to a priest who is competent according to the norm of law without grave inconvenience, those intending to celebrate a true marriage can validly and licitly celebrate it before witnesses alone:
1° in danger of death;
2° outside the danger of death, as long as it is prudently foreseen that such circumstances will continue for a month.
§2. In either case, if another priest, even a non-Catholic one, is able to be present, inasmuch as it is possible he is to be called so that he can bless the marriage, without prejudice for the validity of a marriage in the presence only of the witnesses.
§3. If a marriage was celebrated in the presence only of witnesses, the spouses shall not neglect to receive the blessing of the marriage from a priest as soon as possible.
CIC Canon 1116 §1. If one who, in accordance with the law, is competent to assist, cannot be present or be approached without grave inconvenience, those who intend to enter a true marriage can validly and lawfully contract in the presence of witnesses only:
1° in danger of death;
2° apart from danger of death, provided it is prudently foreseen that this state of affairs will continue for a month.
§2. In either case, if another priest or deacon is at hand who can be present, he must be called upon and, together with the witnesses, be present at the celebration of the marriage, without prejudice to the validity of the marriage in the presence of only the witnesses.