Irishprincess,
This is done in several different ways in the modern era. For those who have civilly contracted their marriages with a state license (or use the common law marriage benefit) and then they go through the courts the family court provides this when the case is adjudicated. It usually has stipulations as to who will receive what in the divorce settlement.
Outside of that realm, for those who practice Christian unions with no specified contract in place, something I would not suggest, when divorce happens the parties often just move along with a verbal divorce statement to one another. I find that to be a problem, especially here on this end when counseling. Why? It is hard to know who is telling the truth and it is hard to know who really initiated the divorce and what it was over and why it happened and who was the victim and who, if either, is innocent.
In a better arranged Christian union, or coalescent union, if the parties will write out a contract, or covenant, and come together based upon that and then should, God forbid, sin destroy the union the man should write out to the woman the settlement paper if he is releasing the woman.
In Judaism the families were under an eldership system (see Ex. 18 and 19) and then in the NT church the saints were under the authority of elders (Titus 1:5; Acts 14:23). There was no such idea of any saint who was without an elder over him or her in that system. In the OT if the man was guilty he could be forced to leave the community, excommunicated, or even stoned if in serious sin. If he refused to release a woman because he was hard hearted he could be dealt with sternly by the leaders who ruled over him. In the NT if the man was guilty he could be excommunicated for being in sin when he refused to repent of any sin. And in the NT if a union was destroyed and it could not be resolved there was always these body of elders over the saints who worked to find out why reconciliation could not be achieved. If one party refused to be reconciled and that one deserted it was that one who was considered to be acting as an unbeliever (1 Cor. 7 desertion texts). Thus, in that case the elders, who were the authority over their disciples, would provide the release settlement or the objective word that person xyz was free from the union because of abc who deserted xyz. The key to both testaments was the authority structure over the lives of the believers.
In today's system people use the courts and their churches and spiritual leaders as objective resources to prepare and guide the divorce papers process.
However, as noted above, some today do not use this biblical system of the civil or spiritual authority structures and they then unite and break bonds with little to no counsel or guidance either way and thus there are no papers or contract or covenants upon entrance or upon exit. It is better, and safer for the woman, if the union is progressively developed and then at some point a covenant made or contracted in some way or another and then if something ever ends it is good for there to be a document that spells out what caused it to end and what is, if any, the settled outcome (property to be distributed, funds to be given, etc). This document would ideally be witnessed by others or by a set of elders who oversee the lives of those people.
Dr. Allen