People can be oversensitive, sure, and I think a lot of social justice/outrage on the internet is completely overblown and mostly the product of very young people forming an identity around being offended or being offended on behalf of other people. Self-righteousness of this kind is a lazy kind of behavior done in the place of being legitimately socially conscious. Reblogging or retweeting something someone said and repeatedly calling them a bigot is not the same thing as actually raising public awareness about an issue or (ideally) working to solve it.
On the other hand, that doesn’t in any way mean there isn’t a massive amount of horrible, hurtful things being said by people everywhere, and that there isn’t a huge amount of the public mindset that has some screwed up ideas of how to treat people. Every thoughtful human being has a moral obligation to combat bigotry and all forms of intolerance, but the way you combat bigotry is extremely important. Doing it incorrectly or for the wrong reasons can make matters worse.
How do you strike the balance? Here’s my rule of thumb: Before saying something, ask yourself: am I doing this to realistically change the mind of this person or others’, or am I doing this to show how much more progressive I am? Nothing’s more venomous than attacking someone simply to assert your place in a subculture, even if it’s a subculture of activism. It’s important to always think pragmatically: if what you’re about to say/type/do seems like it’s more about making your feel better, you should rethink how you’re doing it. Lashing out or even mildly nitpicking can potentially hurt what you’re actually trying to accomplish.
The purpose of calling out hurtful language, etc is to ultimately change human behavior for the better. Anything you do without that goal in mind has the potential to be toxic. Don’t release more hate into the world, it’s the opposite of what you set out to do.