When will the critters appear?

I don't begrudge anyone the right to do to their house what they want but I have to draw the line when it brings the threat of malevolent things to my house.
My neighbors who are foreign born have begun their own trash heap in their back yard along side a shared fence. To date there is a mattress, 4 chairs, various lamps, pots and pans, waste cans, toys, clothing items, discarded electrical items, and a swing set. Today they are out in their backyard cooking up a storm for some get-together. It apparently does not seem to bother them to have all that junk in their yard. Of course it is around the corner from their back porch so they don't really see it when they are out in the yard. I keep watching for rats and other rodents to come scampering out of the ever-growing pile in the backyard.
It brings me to wonder why they don't just put the junk out on the curb for trash pick up or take it to the local landfill for disposal. They have trucks but who knows. The HOA won't do anything because it is not visible from the road. I have to wonder what it will take to get them to clean it up.