I have been hesitant to stay signed in as I don't know if the system admin can tell or not. I don't want to raise questions. Rod? Can they tell?
I don't think it matters whether you are signed in or not. If the web page is open, they can figure that out. If they see that, then decide to find out what goes on over here, they can just access the site. If they see you've made 48 posts from 10 to 10:15, you might be in trouble.
As Geoff points out, they can track what you are doing, if they have the time and inclination. Some do, some don't.
What I do here is have the club log me in automatically. I close my browser and come back, so I'm not usually online for long periods. Not that I think that would fool anyone, but I know for a fact that we don't have the resources to track me. After all, I'd probably be given that job if anyone was.
All of which reminds me of 1984 --
"It's behind the picture." "Yes, it's behind the picture."