All marriages are the same when it comes to divorce. You have to file a divorce petition through a district court, as you would if you had a ceremonial marriage.

If one person believes they are married and the other doesn’t, does that invalidate the common-law marriage?

It is very common to contest the existence of a common-law marriage in a divorce. Often, one party is certain the couple is married and the other is not. Then it becomes a question of proving the necessary elements, which include demonstrating whether others believe you are married and that you have filed paperwork as husband and wife.