To argue, fight, or compete until an agreement or victory has been achieved. My husband and his brother have been fighting it out over who should inherit their parents' estate. The two juggernauts of boxing will fight it out this Saturday for the championship belt.

To argue or fight until a resolution is reached or a victor found. A noun or pronoun can be used between "fight" and "out." My sister and I fought it out until we came up with a workable schedule for each of us to use the car.