Sorry I wasn't very clear. I know what I'm describing so I sometimes have a problem with describing things well enough for people who don't know what I'm talking about already. I also sometimes phrase things badly knowing exactly what I mean and for some reason expecing others to, too. I'll try to clarify.
When I said press the seam open, I was accually refering to the two pieces and pressing them open at the seam so that both strips are laying right side up and ready to repeat the process with the next strip.
You could do additonal quilting if you wanted to, however as long as your 'seam' lines are within the quilting requirements of your particular batting (if you choose to use one) you should be fine with just those. My oldest son has a baby quilt done in this manner with about 6-8 inch strips and no addtional quliting. I was used frequently by both him and his younger brother and held up just fine to the use and the washings. However, you could always do another line of stitching down the middle of your strips if you felt you wanted more. You could even use a decroative stitch and a pretty thread if you wanted to jazz it up a little.
The quilting as you go refers to the fact that you are 'quilting it as you go' about your piecing leaving no quilting to do at the end.
Clear as mud?