It didn't feel appropriate within the boingboing walls to couple a reproductive organ with privacy - somehow, censorial.
Besides, I haven't read much Chandler, if any in fact, and was unaware that it wasn't just "private eye", aka "dick".
There must be some kind of ranking depending on which side of the observation fence you are. The wrong side, you might call them a "Major dick". The side where they're working for you, well, you're going to rank them downwards in the hierarchy, so Private seems appropriate.