No, Santo Domingo does NOT have regular trash pickup just like the US does - it depends where you live. IF you live in a nice area, or along a street where the President or other "important people" pass by every day, they pick it up every night.
If you live in a crappy area, they might come by every 2 weeks or so to pick it up. In these areas, most folks pick a spot, and soon there is a huge pile of neighborhood trash , which begins to smell bad after a day or two, or ten. Yet another reason to check out your prospective new neighborhood, before signing a lease or buying a place.
And one other difference - nobody here, other than some commercial businesses, pays for trash pickup. The local government gets paid by the federal government, the mayor and his cronies take their slice, and whatever is left, is spent to keep trash pickup often enough to avoid news articles. This is best shown by Santo Domingo Oriental, which in most areas has horrible garbage service compared to parts of Santo domingo in the Polygonal.