A disagreement deserves enough privacy for neither person to need a performance for spectators.

An audience changes what an argument is for. Once other people are watching, both sides acquire a second job, which is to look reasonable, and that job competes with the first one of settling anything. Positions harden because retreating in front of witnesses costs more than retreating alone. Privacy here is not secrecy, it is simply the removal of any incentive to win.

When something needs sorting out, choose the setting before you choose the words. A walk, a car, a closed door, anything that lets the other person change their mind without an audience noticing.