Since you are telling Moose to create two functions (a reader and a writer) with the same name, it's no wonder you are having problems!

has priv_attr => (
    accessor => '_priv_attr',
    isa      => 'Str',
);
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has _priv_attr => (
    init_arg => 'priv_attr',
    is       => 'rw',
    isa      => 'Str',
);
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has _priv_attr => (
    is       => 'rw',
    isa      => 'Str',
);
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has _priv_attr => (
    init_arg => undef,
    is       => 'rw',
    isa      => 'Str',
);
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1: Can be initialised using ->new( priv_attr => ... )
2: Can be initialised using ->new( priv_attr => ... )
3: Can be initialised using ->new( _priv_attr => ... )
4: Cannot be initialised from outside the class.
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