Should parents have their teens' Internet passwords?

I hate to say it, but it's not a safe world out there. I think the majority of parents couldn't handle monitoring their teens conversations and such. If you want to keep them safe, then you need to not get mad and interfere if they do something you don't approve of. Otherwise those teens will set up new passwords and accounts that you WON'T get, cause those teens are way more computer savvy than you are.

Always tell them to Think before they Click.  As a parent you should teach them what to share and not to the public.  Give them that responsibility and yes give them their freedom and privacy.  Also, there are tons of software available online that can block abusive websites.  You can also it manually without installing them by following the steps here:

http://www.computeronlinetips.com/internt-tips/block-websites.html

If their given privilege to use the internet has become a bad influence and not constructive to them.  Then you can strict them from using it by putting a password on the internet.

When i was 13, AOL sent me a warning that I used very harsh language via email. It was devastating for me, but it taught me to act safely. Maybe it'd be best not to take passwords, but to catch your teen screw up once, bring to his/her awareness the repercussions, and see a change. 

I once attended a meeting about child safety in which the police officer who was speaking said definitely yes, parents should do everything in their power to protect their teenagers, even reading their diaries! I thought at the time that I would follow his advice as my kids got older. But my oldest is now 14, and I know she is so smart, trustworthy, level-headed and sensible that most of the time I don't feel a need to invade her privacy like that.