I agree with everything Centrarchid said. These things can happen, but there are often differences in what can happen and what will happen.
Many predators can come out during the day. It's not unusual to see coyotes and foxes during the day, and I've occasionally seen raccoons, possums, and skunks. That does not mean they are sick or have rabies or anything like that as some people will tell you. It simply means they are out during the day. Dusk and dawn are really risky times, but your biggest threat is at night. Many are more active at night and they are less likely to be seen. Your dogs should definitely help with that.
Many decades ago when I was young, my parents truly free ranged their chickens. No fences at all except around the garden to keep the chickens out. We were in the middle of pastures and woodlands so the predators were around. Occasionally we would see one. Sometimes we had a dog and sometimes we did not. We would go several years without any kind of predator problem, then a fox or dog would find them and have to be dealt with. Not overly secure set-ups can work well for a long time, but occasionally they can totally fail. You can never tell when a predator will attack or which one it will be.
I do things differently here. I'm still in the middle of pastures and woodlands and I still see coyotes, foxes, and other things during the day. I was free ranging during the day and locking them up securely at night. I only lost two over a three year period, probably to a fox, which was not too bad. When I lost one I would lock the rest in the run for about a month until the predator learned there was not always a free meal running around. My run was big enough I could do that.
Then I went through a rash of people dropping their dogs off out here and lost a bunch. I also suspect a neighbor's new dog may have been part of the problem but I was not home when whatever it was struck. Now they are kept in electric netting during the day, at least until I can build the numbers back up, then I might try free ranging again. The main point of all that is that you can never tell when or if a predator will strike or what predator it will be. In my experience dogs are the worst, but others will have different experiences.
Many of us use a philosophy of having a really predator proof coop and a predator resistant run. My run is not predator proof, but is fairly predator resistant. Part of the top is open so things could fly in. It's possible a raccoon, fox (yes, foxes can climb fences), or something else could climb in. It is high enough that dog or coyote cannot jump in. I have not lost any when they are in this run during the day and locked in the coop at night, even when a known predator had shown interest in my flock. I have not lost any since I started that electric netting in November. That does not mean I won't lose one today. You never know.
I'm not sure what you mean by poultry netting. If it is plastic, it is no protection at all. A dog, coyote, fox, or raccoon can go through that as if it were not there. If it is metal chicken were, it is a deterrent. A dog, coyote, and raccoon, maybe a fox, can still go through it, but it will slow them down and sometimes convince them it's not worth the trouble. A deterrent, not a total stop. I'd still suggest replacing it or maybe just covering it with stronger fencing. The 2"x4" welded wire would be a reasonable one to use, the higher the better. Five feet high will stop most non-climbing things but I've had chickens learn to fly out of a run with 5' high fencing. It's now around 7 to 8' high.
I can’t tell you what is right for your situation and your risk tolerance. We are all different. Good luck!!!