This depends on the state in which the marriage license was issued. Different States have different laws regarding who is supposed to return the marriage license. The most common way of returning marriage licenses is by mail. So for most cases it does not matter if it was done by the couple or the officiant.
That being said, some states and localities within states do require that the marriage license be returned by specified parties. When the couple receives their marriage license, it will come with instructions on its proper completion and return.
If you have further questions, we recommend that you contact the office that issued the marriage license to the couple. Usually this is the County Clerk or some other office within the County Courthouse.
For more information on the marriage license check out our marriage license page.