does it push the value, or just a pointer?

When you're passing around objects, really you're passing around special references. So even if you have two scalars pointing to one struct, it's that same struct you're modifying, regardless of how it's named.

It might be easier to illustrate with an example:

use Class::Struct;
use strict;

struct Test => [ xyz => '$' ];

our $T = new Test(xyz => 123);
print "\$T->xyz starts as: ", $T->xyz, "\n";

no strict 'refs';
my $symbolic = "T";
${$symbolic}->xyz(456);
print "\$T->xyz is now:    ", $T->xyz, "\n";

my %A
$A{t} = $T;
$A{t}->xyz(789);
print "\$T->xyz is now:    ", $T->xyz, "\n";

__END__
$T->xyz starts as: 123
$T->xyz is now:    456
$T->xyz is now:    789
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To get "full" copy effects, you need to make a copy of each of the fields:

my $a = new Test ( xyz => 123 );         # original
my $b = new Test ( xyz => $a->xyz );     # copy of a

$b->xyz(456);
printf "a=%d b=%d\n", $a->xyz, $b->xyz;  # a=123 b=456
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I don't know of an easy way to "deep copy" struct objects, though the Clone module (or Storable's 'dclone' function) may help.