On the side facing you, you will knit with one color and purl with the other color. When you turn your work, reverse the colors with which you knit & purl.
In other words, on the "right" side, knit the blue sts with blue and purl the white with white. When you turn to the "wrong" side, purl the blue sts with blue yarn, and knit the white sts with white yarn.
You will end up with two separate fabrics with the wrong sides facing each other...I would suggest you keep the alternating colors as a border, as otherwise your 2 fabrics wont be connected up the sides of the scarf.
Does that make sense?
It may help you to watch the video on double knitting...I think its on the "advanced techniques" tab....