Ay yia yia.
Western (Latin) and Eastern (Orthodox) Catholics are not in agreement on who the minister of the Sacrament of Holy Matrimony is.
In the Latin Church, the man and woman mutually confer the Sacrament upon each other. In the Eastern Church, the priest is the minister (though, unlike ANY other Sacrament, he must first ask and obtain the permission of the couple to conduct the ceremony).
This difference in theology is not significant between Western and Eastern Catholics, because (either way) the conditions will be fully fulfilled.
But the difference becomes HUGELY important when discussing non-Catholic Christian marriages (protestants are the source of many more East-West discrepancies than filioque). Under Latin theology, assuming both the bride and groom are not Catholic but have received valid Christian Baptism, and they both intend to “do what the Church does” in conferring the Sacrament (even if they don’t properly understand what that means - it is “intent” that matters, not understanding), then the marriage is presumed Sacramentally valid. If this couple later converts to the Catholic Faith (as my wife and I did), the marriage vows are “renewed” and the union receives the blessing of the Church (which is what happened for us).
But the (Latin) Catholic Church recognizes practically ANY marriage as valid (though, possibly, not Sacramental).
The institution of marriage pre-dates the Christian religion, and even the Jewish religion (Noah and his sons had wives, even though Noah considerably pre-dated Abraham and Moses). This is why the Catholic Church accepts the lawful validity of marriage (even if it could not possibly be Sacramental) unless it is PROVEN to be invalid:
Marriage enjoys the favour of law. Consequently, in doubt the validity of a marriage must be upheld until the contrary is proven. [Code of Canon Law, 1060]
Muslims (for example) who are married within their Faith are considered lawfully married by the (Latin) Catholic Church, unless somebody can PROVE otherwise (such as proving that one was not lawfully free to marry). Their marriage is not considered to be a Sacrament (assuming these Muslims are not Baptized), but it is considered to be lawfully valid, and if this couple converted to the Catholic Faith, but did not have their marriage recognized, they would not be considered to be “living in sin.”