Here is a replacement for your module and script showing one way of achieving this.
use parent 'Exporter';
our ($VERSION, @EXPORT, $debug);
$VERSION = '1.0';
@EXPORT = qw($debug);
$debug = 1;
use lib '.';
print "debug = $debug\n";
I've removed a lot of your boilerplate for clarity. I've also adhered to convention and capitalised the module name. Note that filling @EXPORT is generally frowned upon. Much better to use @EXPORT_OK and allow the script to decide which symbols to import.
BTW does Perl have a system variable "debug" that is global and does not need qualification with the package name?
No, but it does have the warnings pragma and $^W which you might consider similar. I'd stick with your own flags unless you have a good reason to do otherwise, though.