So I have been thinking about your idea.
First concern is that the lumpy side of a rag quilt is not comfortable. That is why the smooth side is down, or against the body. If you somehow double the fabric it will be incredibly heavy and take forever to dry.
But if you really want ragged edges on both sides, you can sew long rows together instead of squares. Then alternate sewing the seams one way then the other. You would wind up with alternating ragged rows on both sides.