Releasing need to be right vs. apologizing

I LOVE the idea of releasing the need to be right. I’m currently listening to the Power of Now and he talks about our attachment to our thoughts that makes us sure that being wrong = death. No wonder we don’t want to go there. 🙂

As I toy with the idea, however, my resistance is that if I release the need to be right in a difficult conversation, and opt out of the entire right/wrong system, then the other person — still in that mode of thinking — will still think that I’m wrong, and expect or ask for an apology before they can move on to solutions.

Can you elaborate a little bit on that? Can you just opt out of apologizing if you’re genuinely not sorry because you feel you just had different thoughts and there’s no right/wrong here? Does the full “let me hear you out” phase typically alleviate the other person’s need for an apology?

Or, if you’re completely detached from right/wrong, do you throw them a bone from your high place, like “I’m sorry you feel that way” or “I’m sorry this has been your experience” or “I’m sorry this is how the situation looked to you”. I’ve been trying variations of those, but they feel condescending to me.

Thank you!