I know its late , but I found a decent way to do it without restarting already running process in screen is :-
Attach to your screen with screen -r
Enter Ctrl+A
and then Shift+;
(a small command window will open)
Enter :log on
then detach the current screen by Ctrl+A
and Ctrl+D
And you can see output with tail -f -n 50 screenlog
(where -f tells tail to keep following output and -n 50 shows last 50 lines of output , screenlog is the name of file created in the directory where screen session was inititated )
(Optional - you can set the name of logfile before you turn on log with Ctrl+A
& then Shift+;
and entering name like logfile mylog