I got tired of switching to root just to fix this problem and run screen as my normal user, so I looked it up and found these solutions from Red Hat and Debian. First, as root, I had to change that directory's group to utmp:
Then, the terminal told me that /var/run/screen needed mode 775, so still as root I entered:
and it's been working ever since. Don't forget to run screen as a normal user!