Yes and no. I'll start off by adding that what the students are doing with confidential data is wrong, and should be punished. However, on the one hand, you have publicly-available material on your employees. On the other hand, they're entitled to their private, free time unrelated to their job - and if they choose to spend this online, in private sites, then why should the employer be allowed to be involved? Of course, the easy solution to this is:
1) use a username that doesn't relate to you, personally, on sites like YouTube. Nobody will know who you are.
2) on sites like Facebook, where most people use their real names, simply don't leave your profile open to viewing by "anybody", and don't add potential employers to your friends list!
One final issue is this one of employers actively checking up. That is wrong, and should be disallowed. What if they check my gardening club, or my basketball team, or my local Linux user group? Are they really allowed to go snooping around in my private life and interviewing my childhood friends? Hell no. They're not allowed to do it in real life and they shouldn't be allowed to do it online. As they say, "even the Queen has to use the toilet" - we're all human, and people do silly things IN THEIR OWN PRIVATE TIME, away from clients and bosses. We can't allow that to be taken away.