“The Greeks are famous for their cities and in this they aimed at beauty. The Romans excelled in those things which the Greeks took little interest in such as the building of roads, aqueducts and sewers.” - Strabo

Of all the things Romans had accomplished, the sewers were "the most noteworthy things of all" - Pliny the elder

The Romans loved sewers so much they had a sewer goddess named Cloacina. They also had an understanding of hygiene, if the people are clean they are healthy, if they are healthy they are happy. They were a people obsessed with their ideal of cleanliness. Public bathrooms were probably fairly common amenities (I have never heard anything about how common though). I doubt the romans would take too kindly to someone dumping their shit on the peoples roads, it was meant to go into the peoples holy sewer, people were expected to empty their pots into the sewer.