Here is a drawing of how the plywood is to be placed. Like others have said running the boards horizontally allows each board to be screwed in to more boards reducing the change of racking. I am aware of the nail vs screw debate and in this application the boards won't be holding a load so I am not worried about them shearing.
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As has been stated those are shear walls, and while they do not carry a vertical load, they
are loaded in seismic events and during winds.
The wall with the doors is the most critical in that plan and may require a different nailing schedule than the others. If the building is being inspected, you are going to need approved plans. The architect or engineer who stamps the plans should provide a nailing schedule for each wall. The inspector will inspect to the plans.
While ring shank nails are fairly common, they have recently been found to damage the panel they are driven through, and are sometimes not allowed. Two years ago in CA, I was required to use "screw nails", which are the ones which have twisted flutes along their length. Galvanized vs. non-galvanized, cement coated (not portland, but heat activated glue) vs. plain and smooth vs. ring-shank vs. screw nails are all possibilities.
Bottom line: don't buy the nails until you have the stamped plans in hand, and then get the ones specified.
As others have stated screws are generally not allowed as they are usually brittle and do not have shear strength required for this application.
Use a good nail gun to place these nails. The ones I have seen on most job sites are Hitachi. If you are buying a new one, be sure it will take full 16d common nails. There are a lot of nail guns out there which will not take full round head 3-1/2" long 16 d nails. Some only take clipped heads, and some only take goofy-looking nails with the head not centered on the shank. Some will claim to take 16 d nails, but the ones they use are only 3-1/4" long.
If you get a Simpson Strong-Tie catalog in the back there are shear values for many nail sizes and designs.