§1. After a marriage has been celebrated, the pastor of the place of the celebration or the person who takes his place, even if neither assisted at the marriage, is to note as soon as possible in the marriage register the names of the spouses, the person who assisted, and the witnesses, and the place and date of the celebration of the marriage according to the method prescribed by the conference of bishops or the diocesan bishop.
§2. Whenever a marriage is contracted according to the norm of can. 1116, a priest or deacon, if he was present at the celebration, or otherwise the witnesses in solidum with the contracting parties are bound to inform as soon as possible the pastor or local ordinary about the marriage entered into.
§3. For a marriage contracted with a dispensation from canonical form, the local ordinary who granted the dispensation is to take care that the dispensation and celebration are inscribed in the marriage registers of both the curia and the proper parish of the Catholic party whose pastor conducted the investigation about the free status. The Catholic spouse is bound to notify as soon as possible the same ordinary and pastor about the marriage celebrated and also to indicate the place of the celebration and the public form observed.
Source: Code of Canon Law (Vatican.va)