People need to get over being wrong. But, on a personal level, it's not generally their direct actions that cause this feeling.
Sadly, most societies don't handle "wrongness" well. Even this site is a classic example; where posters are downvoted for wrongness, and this affects their rating, which affects their sense of self worth (and public opinion of them), and pride. It follows that people can't even "risk" being wrong, lest they be downvoted.
This is not only true here; it's true in life more generally. If someone sees you as being wrong, their trust in you (whether they do it on purpose or not) goes down.
What I'm getting at is that change comes from within. You need to stop being afraid of being wrong. You need to actively try and accept people as being wrong, and still trust them (to a sensible degree). You need to be more accepting of peoples mistakes, and let people try things out, and fail, and learn, and continue to do so.
You want to tell someone they are wrong? Tell them.
me: Hey, Joe, that's wrong. Here's why: A, B, C.
Then you pass the same advice on to Joe. Hopefully Joe will pass it on; with any luck Joe will do it back to you one day:
Joe: Silky, you crazy lunatic, you totally got that wrong ...
And just be friendly with people, in general.
I don't think any reasonable person can be upset about accurate advice or comments that are presented in a friendly manner; and if you also do it this way, and still treat the person with respect, and let the same happen with you, I think it can only make for a better system in general.
Personally, this is the advice I adopt globally. I'll be honest; sometimes there are people that get mad at me for it; but bad luck. I do the same with everyone, and I'd expect nothing less done to me. We all benefit.