One user is working on ssh shell. Root wants to monitor user's actions on root's terminal window.
I was thinking about connecting to screen twice. Input and output is required. *cat .bash_history* is not enough.
By Anders' suggestion, ttysnoop is a great utility for this. For something more permanent, you can try out something like rootsh. I have not used this as a default shell, but your sysadmin could easily test how well that works.
You can also look into writing a wrapper around the script utility and setting that wrapper as the default shell.