If script1.pl does not run in the background (does not return before it's actully done), and closes its filehandles when it's done* you should never get this problem.

What operating system do you use?

* update: file-handles are always flushed & closed when your program exit()s, but might remain open or unflushed if your program terminates with certain exceptions or when you call POSIX::_exit() or exec().