Although a few others have posted good information already, I'll toss in my two cents.

And yes, I am aware that I'm not using strict or warnings, and that I should be, but...

Speaking from personal experience, I'd recommend getting into the habit of using these lines of code, which will help you identify problems sooner.

Also, I noticed that you only error checked on just one of the 3 files that you opened, but you didn't print out the $! variable that contains detailed information about why the open command fails. If you add that in every time you open a file, it will help to quickly identify the source of problems if open fails.

Applying the above thoughts along with the stuff from jdporter's and blakew's posts, I'd rewrite your code as the following untested code:

use strict;
use warnings;

open(FILE,"<messages.txt") or die "Unable to open file 'messages.txt':
+  $!\n";
open(my $daniel,">Daniel.txt") or die "Unable to open file 'Daniel.txt
+':  $!\n";
open(my $steffi,">Steffi.txt") or die "Unable to open file 'Steffi.txt
+':  $!\n";

my $fh = "none";
while (<FILE>) {
    chomp;
    if (/^\s*Daniel\s+Lastname\s*$/s) {$fh = $daniel;}
    if (/^\s*Steffi\s+Lastname\s*$/s) {$fh = $steffi;}
    if ($fh ne "none") {print $fh $_;}
}

close($steffi);
close($daniel);
close(FILE);
[download]

Hopefully the code above or from blakew's post will help get you going the right direction.