I usually use the underscore convention to indicate protected methods. For private methods, subrefs held in lexical vars are my prefered way:

my $private_method = sub {
    my ($self) = @_;
    . . . 
};

$self->$private_method();
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If you know how to lexical scope like a fox, you can even keep it private from different parts of your class:

{
    my $very_private_method = sub {
        my ($self) = @_;
        . . . 
    };
    $self->$very_private_method(); # Can call it here
}

# But this is a compile error under "use strict 'vars'"
$self->$very_private_method();
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That's a level of privacy most languages don't have.