How can I start a 'screen' session as non-root user

This is a known problem, if you ssh as root somewhere and then su to become a normal user:

$ ssh root@server
# su -l anthon
$ screen 
Cannot open your terminal '/dev/pts/3' - please check.

It is e.g. described in these posts from 2005

The solution is to directly login as the user you want the screen session to run as.