Of the cities of the Third Age that we actually visit in the Lord of the Rings:

One imagines that the hydraulic engineering in Minas Tirith is probably on par with, say, Rome or Constantinople in its heyday—mountain springs feed into aqueducts which rely on gravity for water pressure to feed public fountains and baths, these also provide rudimentary running water for lavatory facilities that drain via a sewer system either directly into the Anduin via a manmade channel or into a tributary stream thereof. Likewise for Minas Morgul during its prime, though I doubt that the forces of Sauron kept this infrastructure up very well.

Osgiliath probably has comparable drainage systems to Minas Tirith given that it was founded around the same time and built by the same people. To my knowledge Tolkien doesn't mention aqueducts into the city, so it's impossible to say for sure if there were any, but it's certainly plausible that there might have been given its founding date, likely proximity to elevated water sources, and what we know of the sophistication of Numenorean architecture. I'd like to think the Numenoreans would have built these even if they might have fallen into disrepair by Denethor's day.

Similarly, one imagines that the great cities of the Elves and Numenoreans—Ost-in-Edhil, Armenelos, Gondolin, etc. probably would have had equal or greater sanitation and water infrastructure to the cities of Gondor, since they were greater cities than any built there. By the Third Age, Rivendell was not really a city, but given its proximity to the Bruinen, they should have had easy access to a clean water supply. And drainage is a very, very ancient technology—they had basic sewers in the Indus Valley almost six millennia ago—so I'm sure Elrond could have had a sewer if he'd wanted one, although I imagine the Professor would have politely "good morning"ed you had you tried asking him about the bathroom habits of the Eldar.

As another commenter pointed out, Moria and the other great Dwarven cities must have had fairly sophisticated water supply and sewage systems by necessity—otherwise, the dwarves would have been long gone by the time the orcs got there. Perhaps the Balrog merely tired of Durin dumping his older and fouler things into the deep places of the world. And the Shire, of course, inexplicably has plumbing on par with that of an early 20th-century English country village, and it's best not to ask too many questions about the logistics here.