Method calls bypass Exporter, so no, this won't work.

There are several ways to do "private" methods.

1. The standard way is to not bother enforcing private-ness. Just pre-pend an underscore (_) to your method names and let the user consciously make the choice to violate encapsulation.
2. You can put a

   sub _my_private_method {
       my $self = shift;
   
       die "Private method, jerk!"
           unless UNIVERSAL::isa($self, __PACKAGE__);
   
   # ... Continue on here as normal
   }
   [download]

   That's known as a gatekeeper. I haven't seen that a lot. That kind of private allows inheritance to work. The other way, requiring that you must be of that class and that class alone would be to do something like ref $self eq __PACKAGE__, more akin to C++'s private.
3. This applies more to data, but you can change your reference from a hashref to a coderef, effectively making all your object instances closures. This would make all your data private. You could extend it to methods, I suppose. I wouldn't reccomend it because it's extremely non-standard and would be very hard to maintain. There's also a minor speed hit, but it's not that major.

All in all, the best method I've found is #1. Just trust your users. It's the easiest policy to implement.

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