The first thing the webserver wants to see is a header, before it sends your page. If it doesn't then there is a problem. Everything sent out STDOUT goes to the webserver, so either print to STDOUT after the header or print to another file handle.
(I also recommend using the object interface of CGI)

So do this (untested):

#!/usr/bin/perl -w

use strict;
use CGI;
my $q = CGI->new();

open LOG,'>','logfile' or die "can't open logfile\n";
print LOG "Trying to set a cookie\n";

print header( 
        -type   => 'text/plain', 
        -cookie => $q->cookie( -name=>'ident', -value=>'cookie#1')
      ); 

print "Cookie SET";
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#!/usr/bin/perl -w

use strict;
use CGI;
my $q = CGI->new();

my $bar = do_my_cookie_stuff( $q->param('foo') );

print header( 
        -type   => 'text/plain', 
        -cookie => $q->cookie( -name=>'ident', -value=>$bar )
      ); 

print "Cookie SET";
[download]