In short...do not rely on the tap water in the DR for consumption. Most folks here drink bottled water. You can get 5 gallon water bottles from colmados fairly cheap. There is a large water treatment plant in Las Terrenas. We can see it from our house. They do exist. The problem as I understand it is the delivery system is not reliable. Broken pipes, infiltration of dirt and bacteria into the piped water is an issue. So, by the time it gets to your home....no bueno.
We personally harvest rainwater and store it in a large underground cistern. In case of rain shortages we pipe water from the city system into our cistern. Our water is then pumped through a three stage filter. We can drink it from the tap; however, it is an expense some will not want or be able to incur.
As for how locals do water, I see the cisterns on some of the roofs here and I believe they are used to store city provided water. This allows for stored water in case of power outages. Also, city water is not pumped 24/7. There are cases (e.g. drought) whereby water is not provided at all, sometimes for weeks! The roof cistern also provides gravity water pressure. I suspect they drink from bottled water and use the cistern for other needs.
This is my perspective from living in LT.