There are two kinds of disagreement. I’ll call one debate and the other discussion. In a debate, each participant thinks of the other as an opponent, and is trying to defeat him. A discussion is cooperative; the participants share resources to discover what the truth is between their seemingly contradictory ideas.
Have you ever noticed how in a debate, neither participant is ever convinced by the other’s point of view? That’s not a coincidence. People are physically incapable of learning information they don’t like. It happens in the limbic system; you can read more here and here. In a debate, each person’s personal investment in the outcome makes it painful, and thus difficult, for him or her to understand the opponent’s ideas. This mode of discourse is evolutionarily advantageous, because even though you can’t convince the opponent, you can convince the audience that the opponent is stupid. I suspect that this is why in a debate have the urge to insult our opponents directly, giving birth to the ad hominem fallacy.
In a debate, all your opponent cares about is winning. If you want to them to see that their side is false, you have to show them that changing their mind is not the same as losing. This is extremely difficult, and I have almost always failed to do so in practice. Saying so explicitly will not work. They will be offended by the implication that they were refusing to change their mind, and will accuse you of a transgression against an invented rule of debates.
There is only one effective way I have found of convincing the opponent that they can safely agree with you. Find a thing you yourself were wrong about, and happily, respectfully concede that point to them, in a way that causes you to lose face. To this end, it may be effective to arrogantly make points you know are wrong, so that you can concede them later.