Pharisees and all Jews don’t eat unless they wash their hands with fists, following the tradition of the elders. Similarly, coming from the market, they don’t eat unless they bathe themselves...

[Jesus] answered, Leave [the Pharisees] alone. They’re blind guides to the blind. And if the blind guides the blind, they’ll both fall into a pit... Don’t you understand that whatever goes into a man from the outside can’t make him dirty? Because it doesn’t go into his heart but into his stomach and out into the toilet. 

SOJ 16.1 (adapted from Mt 15 and Mk 7)

The Latin term for public toilets was foricae. Such facilities, often attached to bathhouses, were common in cities throughout the Roman Empire. The main room contained long benches along the walls with multiple holes cut into them, through which people could relieve themselves. Enclosed trenches beneath the seats drained waste away from populated areas and out of the city. Vinegar- or water-soaked sponges attached to a stick (tersoria, from the Latin tergere, “to wipe”) were readily available to clean one’s backside. Often water from the adjacent bathhouses was recycled to flush through the sewage passages. Roman systems of water delivery and sewage disposal were engineering marvels of the ancient world.

Communal toilets provided little privacy, but the long robes of the day would have covered the lower half of the body while seated. There was no gender segregation in foricae - it was just one room lined with toilet benches. It's unknown whether women used them alongside men. 

Although constructed to improve public hygiene, the buildings themselves would not have been sanitary by any means. Small windows allowed for minimal ventilation. Reusing tersoria throughout the day would have contributed to the spread of intestinal parasites and infections. It's unlikely that upper class citizens frequented public toilets. Instead, they were constructed in densely populated areas for the lower classes, in order to keep bodily waste out of city streets and markets. 

In contrast to public foricae, the toilet of a private house was called a latrina, which is where the English term “latrine” originates. This was more like today's outhouse - a simple bench constructed over a cesspit. Plumbing to remove waste from individual homes was uncommon, since it could also serve as a portal for pests to enter the residence. Chamber pots were another option, but these required regular emptying. Wealthy families could afford to hire stercorraii, professional excrement removers, while the less affluent performed such undesirable tasks themselves. 

In Greek, the language of the New Testament, the term for a toilet or latrine is aphedron, which literally translates to “away from the seat”. Both Matthew and Mark use this term. Similarly, the Greek word for a pit or ditch is bothynos, derived from the word for “dig deep”. It's also used in this passage, in reference to blind men falling into one. The most common reason for digging a pit in ancient times was to dispose of excrement.

Unsurprisingly, there was nothing more unclean to Jews in the first century than a toilet or cesspit. Ancient markets were notoriously stinky places, and also the location of public toilets. Thus, washing or bathing after visiting one was common practice.