If you take "toughen up" literally, it seems to indicate that the annoying-to-infuriating people who use it have the bizarre premise that they should be able to insult people and just get exactly as much response as they want-- I think what they have in mind is acclaim from people who agree with the insults, and silence or compliance from the people who've been insulted.
However, I suspect it isn't intended to be taken literally. It's a classic bullying move to poke someone literally or metaphorically, and then attack them again for having a normal response to harassment.
I've spent enough time on time online and enough time thinking about what I've seen that I have toughened up to some extent. Damned if I know whether it's entirely an improvement.
And it's taken enough time and been enough work that I can't believe that anyone who who says "toughen up" as a snap command for their own convenience (and, by the way, the "toughen up" contingent are apparently at least somewhat affected by signs of dislike, or they wouldn't be telling people to stop it ) is thinking at all about what they're asking for.
Also, if you take the "toughen up" model seriously, I think it implies that people should be unshameable. Or at least not reflexively shameable, and I'm not either would be an improvement over the human race as now constituted.
You see, much as I despise a lot of what's done with social pressure, I'm uncertain that the average result would be better if people were invulnerable to it. The crowd isn't always wrong.
 In other news, if you spend a lot of time talking about how other people complain too much, you're complaining.
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