There are many strange and intriguing stray people wandering around Cusco. But there are even more stray dogs. Here in Cusco, and in many other cities in Peru (and in Mexico), the "perros callejeros" (lit: "street dogs") are everywhere. It's not something that you see in Western cities, where we have dog-tag laws, council patrols, and "the pounds" (i.e. lost dog homes). But in Latin American cities, you can barely walk one block without passing a canine vagabond or two.
It's not really a huge problem. Most of the dogs are very friendly, very easy-going, and reasonably smart about where to do their business. I've never been attacked by one, or seen one being vicious without due cause. Against all odds, I also haven't yet seen one get run over (although I see close calls about 50 times a day). I guess the main problem is that they're probably quite dirty and full of diseases, but you don't have to (and you shouldn't) touch them, so that's not really an issue either.
Although subsisting in the city isn't much fun for them — you often see them scrounging through garbage — most of the stray dogs look reasonably well-nourished. All up, I think it's nice that the dogs wander around the cities. They're not harming anyone, and they have all the freedom and opportunity in the world. I could think of plenty of worse things than being a dog in Cusco.