Upon running Screen with the -L flag, a file called \'screenlog.0\' is created and all output written to it (which is great), but I will be running several Screen sessions a
Alternatively, there's a way to do it online.
Enter command mode in Screen via Ctrl+a, : and use the logfile command with the name of the file you want as argument, for example:
logfile
logfile whatevernameyoulike.log
Source: Screen man page