I can see bits of both arugments.
Say, in a totally hypothetical situation, if I was trolled on here. I would just roll my eyes and laugh, probably move on. If I had taken offence to being insulted over the internet by someone I've never met in my life, enought to want to kill myself, I would be quite oversensitive, now wouldn't I? If someone was hurting my feelings online, I'd just block them and get over it, problem solved.
Keep in mind though, that younger kids use the internet. They don't have the same amount of resilience as older folk do. I remember getting very hurt emotionally in flame wars when I was younger. What we need is to keep younger kids away from internet forums and such, because older people feel tough behind their monitors and feel the right to troll younger kids.
Now, another matter entirely is cyberbullying that involves people from school and such. Another hypothetical situation: Say you're a kid who always gets bullied, ridiculed and left out at school. Your only escape is through the internet, but the bullying follows you there, too. Here too you can be made an object of ridicule. This, in my opinion, is totally not ok. Because the kid has to see the bullies every single day at school, it's not as though it's just an internet person who can be blocked, and poof! they're gone forever. This kind of bullying has a more understandable effect on someone's emotions, since they have no escape from these people, so I think it's the school's job to intervene with such things.
So I can see both sides. People need to stop being so sensitive in some cases, but people need to stop being needlessly cruel as well.