when you say activities if you mean when your active you send and recieve packets over the internet, and all modems even monitor that. But for the actual sites your visiting? Someone could easily monitor the traffic and see what IP its going from and to, which then theyd find the website. They would need software of some kind though, and to be quite computer savvy id imagine.
Depends on what kind of LAN you're connected to. If you're logged onto a large LAN (such as a company one) then most likely they'll have a log or record of all activities of some sort. All server type operating systems (eg Windows Server 2003) has these types of monitoring abilities. If it's a small home network of a few computers, then most likely no because people usually don't care (unless the system administrator is really nosey) and they'll require Packet Capturing software (eg WireShark) to do so.
And there's no way to tell if someone is monitoring you.
Can a server monitor my activities through LAN connected computer? 
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