Marriage is a covenant between a man and a woman concerning their continuing relationship. Consensual agreement should be involved before a license is granted. True, the Old Testament describes cases of arranged weddings, such as (in particular) that of Rebekah and Isaac. It does appear, however, that Rebekah did have the ability to say "no," but she did choose to go to Isaac, sight unseen, as an act of faith (see Genesis 24:53 - 59, 64 - 67, especially verse 58). Notice that mutual consent was involved on both sides of the marriage at some level.
A primary requirement in making an acceptable marriage before God concerns consent when adults wed each other. Before the sexual union takes place (Genesis 2:24), there has to be a covenant (agreement or license) first. An important consequence of this view is that any religion or culture's ceremony is valid intrinsically in God's sight, whether or not the couples are Christians at the time.
A marriage relationship using a private contract is not sinful based on God's word. It should be witnessed by two or more parties so it is harder for the participants to deny that the wedding occurred (cf. see the principle of Deuteronomy 17:6). It appears that a state-granted marriage license was relatively uncommon until the late nineteenth century or early twentieth century, at least in the United States.
Although a marrying couple can choose to wed by private contract, which was common before modern governments took over the process from churches or the individual families in question. There are no, however, practical advantages gained by using such an irregular procedure. It is usually best to get a license.
Clearly, God delegates to human authorities the power to bind a marriage or annul it when they are not violating His law (cf. the principles found in Romans 13:1 - 7, Matthew 18:18 - 19, 16:19). Hence, it does not matter whether a couple is wed by a church, a secular government, a pagan priest, or a Hindu cleric. They are all considered binding in God's eyes, unless it intrinsically violates His law for some reason (e.g. same sex marriage).
If someone gets married by the government or church, they should petition and ask those same authorities to end their relationship publicly. A couple should patiently wait out the legal process before marrying (and having sex) or divorcing. People should not just whimsically decree a 'beginning' or 'end' to their relationship on their own.
An important passage related to this question concerns Christ's general condemnation of divorce (except for adultery) in Matthew 19:3 - 12, when He was debating the Pharisees' citation of the Old Testament’s relatively easy divorce law. Since divorce involves the nullification of the marriage contract that God approved when a couple got married, it is wrong to break it except under very limited circumstances.
In conclusion, it would be best for couples who wish to be wed to seek to obtain 'the magic piece of paper' or official marriage license. Not only will this fulfill principles in the Bible, it will make it easier for the wife and children to claim and receive any government help should the husband become disabled, killed, or simply abandon the family.