-you can look at the tracking that the computer does on its own by default if you have physical access and your child does not know how to hide data. (browser history), arp cache that kind of things.
- some routers will provide different logging levels that you can look at.
- you can setup network monitors for portions of the network you own, you just can not use the knowledge to access data off of your own segment.
- you can install software on a machine you own
basically these are legal if you have ownership rights to the machines or physical hardware. The problem is people get carried away and cross a legal line. For example: it is legal to install a keylogger on a machine you own. it would log a username and password, you read the username and password. All of this is legal.
Now, if you were to use the username and password and log into the external account. It would generally be unlawful. (exception: you own the server or had been granted permission by the server owner)
anyway, laws are different all around the world. It is a subject people just avoid here.