Your Laptop, Your Network
No, they cannot see what you're doing on your laptop when you're on your network, because you're using your machine and accessing the internet through an ISP that's not theirs. You're right to suspect that monitoring you in this situation would be illegal.
When you're on their network, however, the game changes. They can monitor your internet traffic. If you're browsing by way of HTTPS, they can see what domains you visit,
https://www.website.com for example, but not that you ventured to
https://www.website.com/questions. In this case, the
/questions element would remain hidden. Assume that your activity is being monitored while you're on their network, even when you're using your own computer.
I know this is beyond the purview of your question, but note that they can monitor far more when you're using one of their computers. They can, for instance, install a keylogger, have a program take snapshots of the screen at random times, etc. For this case, I invite you to explore the following thread for additional information: Can my employer see what I do on the internet when I am connected to the company network?