There are some people who are quick to make others feel bad by calling the other person out, and then there are other people who always feel bad calling others out on their mistakes. I've always been the latter. I don't ever want to make anyone feel bad because it was just that - a mistake. However, as you get older you start to realize that you can call people out in a respectful way, and it's totally okay to do so.
A lot of times, calling people out is misconstrued as throwing someone under the bus which has a very bad connotation. Usually the person being called out feels attacked and feels betrayed by the person who is calling them out, which I think is why people are so quick to get upset when their mistake is pointed out. However, you can definitely call people out without going out of your way to make them look bad. Sometimes by pointing out a mistake, you're not trying to make the person look bad, but you're trying to make the overall situation better. It's okay to call someone out when their mistake is harmful to a situation or can create more problems than it can do good.
You can completely call someone out while still doing it respectfully. Instead of pointing fingers and making the person feel bad for their mistake in front of other people, you can pose it like a question by saying something along the lines of "I believe this may have been a mistake, was it supposed to be this other way?" or you can put it on you by saying something like "I thought it was supposed to be this way" and that way you're still getting the point across without making them feel bad.
In any situation for calling another person out, however, you need to make sure you're standing your ground if by not calling the person out you're shooting yourself in the foot. It's okay for another person to make mistakes, but when you don't speak up when you catch that mistake, you're not only hurting yourself or other around you, but could cause a chain reaction of negative.
It's hard to call someone out when you're in a situation of them being in a more authoritative position than you. That's when it can get really tricky, which is why you should always do so respectfully. It's okay to get nervous when speaking up because you don't want to undermine someone - whether they have more authority than you.
Speaking up is a hard thing to do, especially when it can be misconstrued as calling someone out in a negative light. It's never okay to intentionally hurt someone else by throwing them under the bus and making them look bad. There are times, however, that you have to call someone out and you need to know that it's totally okay to do so. Just learn how to do it respectfully.