For instance, I know that "being on the street" means being somewhere on the streets, while "being in the street" means being in the streets where cars are being driven. However, I am not sure if the same thing can be said with the verb "pee". So is "peeing on the street" equivalent to "peeing anywhere on the street" and "peeing in the street" equivalent to "peeing in the middle of the street"?
"Peeing on the streets" describes the consequence of the action (urinating onto the surface of the street).
"Peeing in the streets" (or, more commonly, "peeing in the street") describes the location of the action (standing in the street while urinating). Note that the specific location (side, middle, right lane...) is not explicitly understood. You can assume "the middle of the street," though, because the idiom can refer to any location on a street, not just the physical middle.