Sounds like you built a great coop that will last! Different chemicals used now, and much safer all around. If you are really concerned, you can seal it - stain or paint - depending on what you prefer. You'll have a number of choices to choose from at the big box stores. Typically, with most of the sealers, you actually need to let the wood dry for up to 2 months (read the label on the different products as they will give you the specific application guidelines) before you apply it.
As pointed out, sawdust is the only thing you want to watch out for, and you are supposed to use pressure treated approved fasteners that will not erode in PT wood. When I built my enclosed run, I used stainless steel in some areas, and then the approved fasteners in others depending on the fastening need.
The much scarier stuff is when people use the old school railroad ties that had the creosote in them. The modern PT wood available today, don't sweat it. In 20 years when you're coop still looks great and is solid, you'll be glad you did it!