I do not believe that they would record your activity by putting a specialized software on your computer unless you are working on a top-secret project or something.
They may use key loggers to record key strokes on the system but this is very unlikely and unethical and this itself is a cost and a threat when using third-party software to monitor employees.
When you are connected to the company network, VPN or domain then yes, it is possible that they can monitor all activity.
When you are not connected to the company's network/VPN/domain, then you are not transmitting/receiving any data through them.
Tracking Cookies come into place when you visit websites, and those websites place a cookie on your system to track your activity.
Normally, if a key logger is placed on your system then you should be able to see it under the processes or through Control Panel - Add/Remove Programs etc, but like i said this is unethical and i hardly believe any company would use such a system.
Companies are normally concerned about their data, to protect that, they put their own data behind firewalls and secures employees by asking them to connect to VPN (where data gets encrypted) and that's for their protection.
When they offer a job, they put all the terms in the contract and if there are any concerns where you can be held liable, they they clarify it.
Hope this clarifies.