In Washington, the process works like this:
1. Couple fills out a marriage license application from one of the state's counties. The parties must either sign in the presence of a county clerk or deputy county clerk (who, in this state, automatically have powers of a notary) or a notary public. The idea here is to verify the identities of the parties and to take two jurats (legal oaths) required by the state: the parties "swear or affirm" that they meet the legal requirements to be married in Washington State; and that they either have no sexually transmissible diseases or, if they do, that said condition is known to the other party.
2. Couple files marriage license application with the county from which it was obtained. They get the marriage license itself, which becomes valid on the third day after it was obtained. (Ie pick it up on a Monday and the earliest the couple can get married is Wednesday. I've heard an amusing story as to why this is, but that can wait.)
3. For 30 days starting the day the license becomes valid, the couple may stand anywhere in Washington State, before any person authorized to conduct a marriage and in the presence of at least two witnesses, and state their intention to live together as a married couple. (Washington requires no more ceremony than that.) The couple and the two witnesses sign the marriage license in the presence of the officiant, who then signs also. Technically speaking, at that moment, the officiant is acting as a notary public who has taken the jurat of the couple being married. No notary public or other clerk of the court -- except the officiant -- is necessary. Under state law, an officiant may be a superior court judge, active or retired, or a member of the clergy. If the license is not used in the time period given, it becomes invalid and the couple will have to file a new application.
4. The marriage is now legal. Officiant takes license and has 30 days, starting from the day it was signed, to file it with the county from which it was obtained.
It is my understanding that most states operate in a similar way. I know for certain that California, Oregon, Idaho and Nevada do.
Another issue would be addressed by such a change in the laws, one that I had overlooked: the problem of defining "clergy." Past court rulings have declared it to be unconstitutional for any law to define who is and who is not clergy of any religion. It would be perfectly legal to claim self-ordination in your own church and start doing marriages. (My own credentials come through the Universal Life Church, which is pretty close to that.) Washington gets around that through RCW 26.04.060:
A marriage solemnized before any person professing to be a minister or a priest of any religious denomination in this state or professing to be an authorized officer thereof, is not void, nor shall the validity thereof be in any way affected on account of any want of power or authority in such person, if such marriage be consummated with a belief on the part of the persons so married, or either of them, that they have been lawfully joined in marriage.
So as long as you
claim to be clergy, whether or not you really are, the marriage is valid. This change in the law would make it much easier to prevent fraud and other abuses.