Among the healthy and mature, there's no right "not to be offended"; not for men, and not for women. There is 100% equality here.
Such offense is subjective; every possible attempt to minimize it by law boils down to an unworthy suppression of freedom, something that is unhealthy for society no matter how you go about it.
Even when a particular mode of speech, or some consensual/personal action, is pretty much uniformly despised, it's far better to know who says, and therefore has motivation to say, or does, these terribly offensive things, than it is for society to repress these people and then jump up in stark surprise when they move from unseen and unheard to resentful action as a means of kicking back against said repression.
Speech, in many cases, serves as a moderately effective safety valve. You never want to close such a valve and walk away. Because you get this.
If something you look at offends you, look away. If something you hear offends you, stop listening. If something people do offends you, don't participate. Your subjective feelings of offense can never rise to the relevance required to legitimately regulate the behavior of others.
Until something breaks your bones, damages your property/finances, or impugns your reputation, or these things similarly directly affect those for whom you perform the role of parent or guardian, the correct action is to turn to managing your own sensibilities -- rather than trying to control other people's actions.
Now, as to the immature and incompetent, in particular, children: Parents and guardians have a dual responsibility here. In order to be able to execute that responsibility, your home should be a safe haven in the sense of you being able to completely control who, and what information, gets in, and when they get in, and when they must leave. Society owes it to you to see to it that this capacity is readily available to you. Your home should indeed be your castle. To the extent it isn't, society has either failed you, or you have failed your charges. Schools and/or any other situation requiring attendence must likewise be supportive and safe, or society has lost its legitimate right to force your children to attend.
On the one hand, it is your responsibility to see to it that your charges are not bullying, generally or specifically being an asshole to others. You are responsible for inculcating the understanding that immature and/or insufficiently abled minds can be taken to, and beyond, the brink by bullying, and then you must see to it that this understanding translates into reasonable behavior by your charges (which, by the way, will work to reduce many types of essentially pointless trolling later on.)
On the other, it is also your responsibility to see to it that your charges are not being bullied. You should know where your charges hang out, who they hang with, what the environment is like, and you should step in when that environment, in your estimation, becomes unhealthy. Stepping in may involve a note to someone else's parent or guardian, removing your charge from the harmful environment, or simply providing sufficient perspective so that the behavior is seen in the light of failure of the perpetrator, rather than any kind of lessening of the value or self-image of the target.
If your charge cannot be taught to healthily handle the speech, displays, or consensual actions of others, then it is your job to see to it that they are not exposed to those things. It is not society's responsibility to turn the entire planet into a padded room for your charge. If you need a padded room, you should build one of your own.
For every story I have heard so far of horrible consequences to bullying, my reaction has been "Where were the parents during all this?"
And I have to ask: If your charges are not being raised with healthy self-images and a strong sense of self, what the fuck are you doing? And why are you doing it? Why are you such a totally shitty parent or guardian? And why do you expect the rest of us to compensate for your failures?
Again, these people's abject failure at parenting does not rise to the level of telling everyone else they can't call someone something when, in fact, it is pretty apparent that something is called for.
Having said that, most online forums and comment sections are not public operations. They're private. And in that role, they have both the power and the right to monitor and control the content and activity on that forum. If you invite people to spend time in what pretty much amounts to an environment you created, then you'd better tell them up front what the limits, if any, are for that environment, and see to it that you are accurate about it. If (points at facebook) you allow your operation to get too large or otherwise out of your effective control, I really don't see how that absolves you in any way from being absolutely clear to all participants that you are not, in fact, able to guarantee any particular kind of environment or control what is going on. And no, burying such things in a veritable tar pit of legalese doesn't suffice. Be plain; be clear; let no user into your "thing" without a road sign that says "alligators!" or whatever else it needs to say. Because it does need to be said.
The very idea of freedom requires a concomitant effort to ensure a competent citizenry. If you create a nation of pearl-clutchers, you will have created an environment where repression is the always go-to of the regulators. To some extent, this is already happening, particularly in nations like England, which has pretty much fallen off the wagon of sanity and is busily engaged in chewing its own tail off. It would be lovely if the USA didn't follow them any (or at least much) further down that road.