Yes, I very much believe parents should covertly monitor the activities of their children. The world is a very dangerous place for children. There are predators, criminals, drug dealers, and bullies around our schools. Sometimes the biggest danger to children is peer pressure, which comes at them while in school and online. It is very easy for a child to get caught up in the wrong crowd, and be terrified of not being "cool".
Yes, it is a good idea for parents to covertly monitor the activities of their children. Parents have a duty to keep their children safe from harm and monitoring their activities can help to accomplish this. In today's society it is easy for children to get into trouble, so parents should monitor them to do everything possible to keep them safe.
Parents need to walk a fine balance of monitoring their kids (in whatever way) and giving them trust. I think once a child has proven themselves to be dishonest and making poor decisions, it is the parent's responsibility to monitor them to make sure they are on the right path.
I do not think covert monitoring works as well as monitoring where you tell your child that they are being monitored. Covertly monitoring is just trying to catch them in the act of something so you can get them in trouble, where as honest monitoring is something that's done preventatively and is much more respectable.
My mother was extremely curios and controlling to the point were she breached my privacy in ways which ruined my trust and respect for her to this day. Once trust is gone, it is gone for a long time. Good intentions don't mean anything, and don't excuse despicable behavior. It is a parents duty to teach children to make the right choices. Teaching them to invade privacy by example is not the right choice.