Many older cities treat stormwater before it is discharged. Quoting Kate Asher in Anatomy of a City (2005), "New York is one of only 800 or so cities in the [United States] that rely on what's known as a "combined sewer system"—one that mixes storm water with wastewater, and sends them both on to the same treatment plant." Page 174.
Combined sewer systems have "combined sewer outfalls" -- i.e., the ends of pipes that discharge treated water into waterways. These outfalls are well documented, have publicly visible signage and each have a permit. During times of high volumes of rain when treatment plants can't handle it all, "combined sewer overflow" (CSO) events release untreated stormwater AND untreated sewage directly into waterways. See https://www3.epa.gov/region1/eco/uep/cso.html
As an add-on, most stormwater flows into storm drains and directly into streams without treatment. In combined sewer systems, CSOs happen periodically during storms. Separate storm systems avoid this problem. However these flow nearly directly into streams, so if someone poured paint or oil down a storm drain, it would flow untreated into a stream, and eventually out to the ocean.
Some places, particularly newer developments in environmentally friendly areas, have some types of stormwater facilities for minimal treatment. These can be things like catchment basins that separate out gross solids (like litter)., oil-grit filters that trap some of the oil and other fluids leaked by cars. Even stormwater management ponds can count as stormwater treatment. Aquatic plants can cut some of the pollution. Some settles to the bottom of the pond, where it may or may not be eventually dredged out if the pond is being maintained by a responsible business owner or municipality.
An important aspect of stormwater management is simply slowing the water down so it hits the streams with less intensity, and causes less erosion and sediment in streams. This sediment severely affects wildlife, carries pollution, so a large goal in stormwater systems is to slow down and spread out stormwater. But treating this water before it re-enters streams is not common.
"Stormwater Facilities" or "Stormwater BMPs" (Best Management Practices) are terms used to describe these methods of minimizing stormwater's environmental impact. https://en.wikipedia.org/wiki/Best_management_practice_for_water_pollution#Stormwater_management_BMPs