I will attempt to share with you a simplified explanation of what sizing is, just enough to give you an idea why it is important to learn of it and how sizing can affect your preference for wetting your paper for stretching. We do not need to go too deep into it for our purpose. But should you wish for a more comprehensive explanation, you'll find a lot of more technical information available online.
So what is sizing? Sizing is another term for the addition of gelatin to water color paper. Gelatin makes the paper less absorbent and this is what prevents paint from just spreading on the paper uncontrollably. Picture what happens when you dab your brush loaded with water and paint on rag or tissue paper. It spreads, right? Well, the same thing would happen to paper, if it is not sized. This would help explain why there are some papers that look very nice, like handmade paper, but you can't seem to gain any control when you try painting watercolor on them. Handmade papers for crafting projects are often unsized.
Different paper manufacturers and brands offer different sizing options but generally, you can say that some gelatin sizing is incorporated into the paper pulp or mixture before it gets pressed. Additional sizing is applied as coats on the paper's outside layers (front and back surfaces).
Now why should this knowledge affect how you would wet your paper? Just this. We know sizing makes the paper less absorbent. The more sizing, the less running of paint. If you want more absorbency, you lessen the sizing. Conversely, if you want less absorbency, you keep the sizing. The presence of sizing also makes your watercolor pigments look more brilliant or intense as it keeps most of your paint on the paper's surface. You lessen the surface sizing, you increase the permeability of the paper. With less or absent surface sizing, your colors will sink deeper into the paper and will bond more with the paper's fibers. Some mistake this for paint disappearing or pigment bleaching (colors not so lightfast) but the paint is not really gone, it has just gone into hiding deeper among the paper fibers hence the lightening of paint applications as it dries.
We wet the paper to expand the fibers and then staple or tape it while in this expanded state so the paper will have an anchor at its edges to pull on as it dries. This is what allows it to shrink flat (when you do it right) hence the term stretching your watercolor paper.
Some soak the paper completely for several minutes, not just to ensure all of the fibers get wet in preparation for stretching, but also to lessen the sizing of the paper. The correct timing for the soaking takes practice. You will not want to soak it too long because sometimes the sizing have a tendency to coagulate in spots if the paper is left too long by itself. This is the reason why some may find those irregular and slightly yellowish spots on their stretched paper that do not go away when the paper dries. For some reason, these spots also wouldn't take in color as well as the other clear areas which is why I surmised, it must be sizing. These spots will resist any color placed on it. They may ruin your painting. The too long soaking is one explanation for it. Another is that the spots may be the result of accidental drops of pure sizing as the manufacturer applies surface sizing on the paper. Accidents can happen, right? Or, you may have gotten an old stock paper with the sizing already going stale. Whatever the reason, I find that spots are less likely to happen when I wet by running water than by soaking.
For the running water method, you can just put the paper under tap or running water. Make sure all surfaces get wet (front and back) and continually move the paper around. Don't just point the water jet on one area as this may also produce the same sizing spots on your stretched paper. For bigger paper sizes, use the shower for more manueverability. This takes less of the sizing off than what complete submersion does. Again, more sizing present, the less tendency for paint to spread uncontrollably. Your colors also will tend to be more brilliant per application as you will have most of the pigments sitting on or closer to the surface of the paper because of the sizing's effect on permeability.
But my personal choice is putting the paper under tap water and removing most of the sizing by mechanically running my hands over the surfaces of the paper. Your preference would depend on how you want your paper to behave. I like letting more pigments sink into the paper fibers. It allows me to play or take advantage of transparent watercolor layering. It also seems to add brilliance to layered colors (in my opinion and observation only), when you have more it sunk into the fibers. This method also offers the least chance for the sizing to coagulate as you let it run off the paper when you do the sweeping motion. No chance for displaced sizing to stay in one place and make spots. You have to develop a gentle touch though when you run your hand over the surfaces or you could end up agitating and damaging the top fibers. Lessening the sizing works for me because I like to blend a lot for the first part of my painting process. For detail work, which I prefer to do on the last stages of my painting, I mostly use dry brush so even with less sizing, I still maintain enough control over my paint application. By this method also, I remove mostly just the outside sizing. Remember, there is still sizing mixed in with the paper pulp or mixture itself so the sizing is not gone completely. It is possible to maintain still a lot of control with your paint application.
If you want the sizing to be intact, you can always just staple the dry paper onto a board (for 140lb) or use a heavier paper to do away with stretching altogether. If you want to staple or tape in place, you can just wet the paper on top of the board with a wet sponge prior to anchoring. Take care not to agitate the surface too much with the sponge. An alternative is to use a water spritzer or mister to wet the paper so you don't touch the surface of the paper with anything. When the paper dries, the sizing would still be there.
I wish to include here a contribution by Stan Hughes regarding additonal paper options. "There is a 200 lb cold press paper produced by Saunders Waterford. I have used it for years. Easy to work with and does not need stretching. Not as expensive as 300 lb but more durable than 140 lb. "
200 lb paper seems to be a good option to try for when you like to work on paper with intact surface sizing. Thank you very much Stan for the info.
Troubleshooting watercolor paper discoloration
Affordable and easy solution for unprotected wooden boards used for stretching watercolor paper
Additional protection you can do yourself for your framed watercolor painting