Thank you, praised monks, for your valuable input!

I have solved the issue by changing

our @EXPORT = qw( %debug $d_area dsay );
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to

our @EXPORT = qw( %debug *d_area dsay );
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in Dsay.pm.

I now can use debugging code like

sub one {
    local $d_area = "one";
    dsay "debugging output from one";
}
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without the need of qualifying $Dsay::d_area anymore.

Debugging output for the specified area stops as soon as the localized $d_area value drops out of scope.
This is exactly what I wanted.

Thank you for sharing your knowledge!