Generally speaking, you will want to call the _topsecretsort sub as a method, not as a function. The primary reason would be that _topsecretsort may need access to $self (perhaps so that it can call further methods), and even if the current implementation of _topsecretsort does not require access to $self, future rewrites of it may.

That said, I'd hesitate to call _topsecretsort a "private method". It is not a private method in the same sense that programming languages with true privacy would understand it. _topsecretsort can be called and overridden by superclasses, and by code entirely external to the class. I think the best term for this sort of sub would be an "undocumented method", not a "private method".

(In fact, getting back to whether the sub should be called as a function or as a method... the one advantage of calling the sub as a function instead of a method, would be that it would prevent superclasses from accidentally or deliberately overriding it.)

The best way to approximate true private methods in Perl is to use coderefs:

my $_topsecretsort = sub {
   my $self = shift;
   my ($x, $y) = @_;
   return 42;
};

sub count_sorted_foobars {
   my $self = shift;
   my @results = $self->$_topsecretsort('foo', 'bar');
   return scalar(@results);
}
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In Moops I've even created a shortcut for this pattern:

use Moops;

class FooBar {
    method my $_topsecretsort ($x, $y) {
        return 42;
    }
    method count_sorted_foobars () {
        my @results = $self->$_topsecretsort('foo', 'bar');
        return scalar(@results);
    }
}

say FooBar->new->count_sorted_foobars;
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