Hee. . . I don't think it's anatomical.
The directions are
to lift the wrap onto the needle and work it together with the stitch it was around. I really have no clue what the private side of a stitch is.
I've heard the term private side for the wrong side of a garment, but a stitch?
When I pick up a wrap, I slip the stitch to the right point and then lift the wrap with the left tip and put it on the needle and slide them both back. I suspect the 'private side' thing is keeping the wrap either before or after the stitch itself, whichever is less visible.