I believe that the Othodox CHurches do not hold this. For them, it is the priest.
I remember reading somewhere (in a priest's tricky online Catholic quiz, I think) that, since the ministers of the sacrament of marriage are the bride and groom, you could technically marry yourselves on a desert island without a priest or deacon. (You shouldn't, but you could.) Since that's true, then the consent of the spouses is "indispensable."