I live in Washington state and we have many Cougars as they are called out here. We used to have a hound season for cougars, but the animal lovers pushed through a ballot measure that outlawed hunting cougars with hounds. Anyone can get a cougar permit to hunt cougars but it is very tough without hounds, most cougars that are shot are harvested by deer or elk hunters that come across them during deer or elk season. Cougars are very wary and it is unusual to see them. I spend a lot of time outdoors hunting, fishing, and hiking and I have seen one cougar in all that time and that was one by the road when I was in my truck.
I would be concerned if I saw a cougar and it was unafraid, they always say that if a cougar appears to be stalking you you should raise your arms above your head to appear bigger, make noise and stand your ground. Of course if your armed, you have other options.
In spite of the large number of cougars we have out here, attacks on humans are rare, at least here in Washington, in CA they have had more attacks and a couple of people have been killed. But here in WA there haven't been any fatal cougar attacks in many years if ever. Cougars, when they go after humans tend to go after women and children as they are smaller.
We do get problem cougars and bears that wander into the suburbs, usually old or young animals that start killing the local pets, the Game Department will get called and they'll come in and remove the animal.