I would assume it stays local for the most part. However you need to have internet access as you must supply your netacademy or other Cisco credentials in order to start and use packet tracer after so much time has passed.
I dont think there is a "central server" where anything goes. Also not sure what you mean by activity. Activity as in configuring devices or just opening a lab.
Labs are freely available on the internet provided by several vendors if you need them as well.
-David
