Suppose that taking offense is something that you do in response to something that you think lowers the status of either you, your group, or something that you hold dear. Now there are two extremes as to how to respond to possibly offensive things, both of them wrong:
1. Shun anything that could cause offense to anyone. This is wrong because it would unreasonably restrict communication, because people can perceive something to lower their status when it doesn't, and it would allow status grabs where groups decide what you're allowed to say by arbitrarily defining things they don't like as "offensive".
2. Say that being offended is your own fault, and maintain that anyone should be free to say anything they like, regardless of whether it offends someone. This is wrong because some words or expressions, when used in certain contexts, do genuinely attack people or groups by lowering their perceived status. Social norms that were offense-blind would strenghten existing status differences.
The challenge is finding social norms that are balanced between these two. E.g. slurs against a genuinely high-status group are more acceptable than slurs against a low-status group, but the status of a social group is constantly changing and different people have different ideas of a social group's status.
Suppose that person X says something that offends person Y, but X thinks what she said was fine. This could be because of two reasons:
A. X does think that her words might lower the status of Y's group, but Y's group does deserve to be taken down a notch.
B. X genuinely did not intend her words to lower the status of Y's group, and thinks that most listeners wouldn't think that she did.
In the case of e.g. offensive drawings of Mohammed, either or both reasons could be true. X might think that all religions should be mocked because religion is for idiots (A). Alternatively, X might think that all religions get mocked but that doesn't make them less valuable and important (B). Or X might think that yes, this does take lower a religion's status, but only because that religion's adherents have given it such a high status to start with. If they didn't (in X's mind) unreasonably privilege their own beliefs, they'd have no reason to be offended by an assertion that their religion is of no higher status than any other belief (A&B).
Online, it often seems to be the case that people fall prey to the illusion of transparency. If you're with a group of friends who know you well, then it can be safe to say potentially offensive things, since your friends either share your opinion of the other group's status, or know that you don't mean any harm. But if you speak online, where not everybody does know you or your friends, then they cannot know what your intentions were.
People who go with the general strategy of shunning anything offensive tend to presume that reason A is always (or most often) the only right one, while people who think everyone should be free to say anything tend to presume that reason B is always (or most often) the only right one. I don't think either assumption is right.