Could you explain a bit more what your exact situation is? The question you asked is unclear to me.
I understand what the wrong side is, however I'm not sure how to start knitting the wrong side.
if the wrong side is a purl, then do I start knitting on the purl?
if the wrong side is a knit, then do I start knitting on the knit?
What is the pattern stitch that you are using? Besides that you probably need to give the row before where your question arises and then tell us what the row you are on tells you to do.
It depends on the result you want to get how you handle each stitch on the wrong side.
Are you on the very first row of the shawl, or have you done some part of it and are starting another part?