To stretch watercolor paper, you are supposed to soak the paper in water, then squeegee it onto a board and staple all around the edges. Allow the paper to dry before you paint on it. The paper is supposed to stretch while wet and then once mounted and stapled to the board, it is supposed to dry tight as a drum. When stretching 140 lb. paper once stretched, it is supposed to act like 300 lb watercolor paper. Personally, I have not found the need to actually stretch paper. If you want to paint on 300 lb. watercolor paper, buy it.
When I use either 140 lb. or 300 lb. watercolor paper, I tape it to a board using regular masking tape. I do tend to often paint very wet into wet watercolor paintings and I have not had a problem using this method. I tape around all four sides completely with masking tape. Most of the time the paper dries completely flat when the painting is done. Next week, I will have a tip on what to do with warped watercolor paintings.