I modified your module slightly as follows:

package my_module;

$VERSION = '1.00';

use strict;

use Data::Dumper;

our $gMessage;

sub import {
    ($gMessage) = @_;
    print Dumper(\@_);
    print "Yep, I'm in here ... \n\n";
}

&my_sub();

sub my_sub {
    print "Yep, I'm in here, too ... \n\n";
    print $gMessage;
}

1;
[download]

And this is the resulting output:

$ ./testmymodule.pl 
Yep, I'm in here, too ... 

$VAR1 = [
          'my_module',
          'xyz'
        ];
Yep, I'm in here ...
[download]

That should clarify why you do not see 'xyz' being printed.