My Xbox 360 says
Your console can't conncect to Xbox Live. Do you want to exit the session and test connections?. I don't have Xbox Live and don't want it. I just want to sign in on my profile and play games. Can anyone help please?
Create a local profile. I have couple of them, for players that I play on split screen. Such account isn't sync with Live.
You can create such accunt via accounts from screen menu.