It occurred to me that these ancient cities would have needed a way to handle waste. Modern sewage systems are closed, and in the first world at least, do not pump raw sewage into waterways. The Romans were known for their innovation in road building, did this innovation extend to plumbing and sewage?
Also, did ancient sewage go into rivers/the ocean, and how did it get there? What were the conditions of the houses and streets? Were there open sewers or were there 'pipelines' of some sort? What level of awareness of hygiene was there during these times?
(Also: information regarding sewage in any major city during the same era would be equally interesting.)
The ancient Roman sewer system is not so drastically different from our own modern systems from a functional standpoint minus some obvious advancements.
The main artery for the modern Roman sewer system was the Cloaca Maxima, or "Greatest Sewer" Initially it was an uncovered canal dug out to drain the marshes around Rome and dispose of waste and would flow into the nearby Tiber River. Its estimated that other canals existed anywhere from 800-500 BC but this would be the first major Sewer line. Also Rome is very hilly, without adequate drainage you would have bad foundation for building, and the loss of soil on which to farm.
The first Aqueducts would start appearing around 315 BC to bring fresh water from from far away sources. Aqueducts were constructed and worked into the main grid to service latrines, baths, fountains and pools and eventually private households, all eventually draining into the Cloaca Maxima and out into the Tiber. If you've never read up on Aqueducts, they are an amazing feat of engineering, using gravity to bring water instead of modern pumps we use today. Many of them still survive, without the amount of quarrying that took place through the Dark Ages, some would still probably be almost naturally functional. The Trevi fountain last time I checked In Rome is actually still fed by a restored natural aquaduct.
Anyways, The Cloaca Maxima would be expanded and covered around 33 BC, while the Aquaduct system would keep expanding until around end of the empire.
The Cloaca Maxima is actually still functional some 2500 years later draining the Forum and a few other places. There are still Ancient Roman arches covering the Maxima underneath the modern city, although much of it has fallen into disrepair and has, or is collapsing.
As far as Hygiene? Well, it's not to our modern standards, lacking scientific knowledge of how diseases spread. There are accounts of public latrines where people would wipe thier bottoms with a sponge and dip it in water or vinegar afterwards. Imagine using the sponge at the end of the day...
Baths could also be problematic, not understanding water born bacteria transmission. If you were in a heated public bath that had been practically been stagnant all day, various people with possible open wounds bathing and leaking bacteria into the heated stagnant water, well, you see where that could go. Infection was often a death sentence lacking modern medicine and knowledge in the ancient world. And many times, the cure was often worse than the ailment, lacking knowledge on blood and airborn pathogens.
I am not well versed in Athenian sewer systems and drainage, but there are some sources post the conquer of Greece by the Romans that suggest the greeks admired Rome's sewer system.
I don't have the expertise to answer your question, but I feel it's worth pointing out the disparity between the places you're asking about. Athens, while a decent-sized urban centre for much of Antiquity, fit within the parameters of cities of the time. Athens itself may have had up to 40,000 inhabitants at its largest Classical size. When Rome was at its height, the second largest city of Italy (Capua) was probably of a similar size as Athens. The city of Rome, with a population of around one million by the 1st century AD, is a historical anomaly of staggering proportions.
Needless to say, the problem of getting rid of waste was vastly greater in Rome than it was in Athens. As far as I know, Classical Athens had no sewage system and no infrastructure to get rid of waste. Squalor was a known cause of disease and discomfort (even if the exact link between filth and disease was not understood), so Athens had public fountains since at least the 6th century BC, but there was no architecture in place to take care of waste (and not enough running water to make it work). Only at times of extreme pressure would this cause problems. Rome obviously did not have the luxury of being so lax about sewage, and it is at Rome that you'll find the most innovative solutions to the downsides of concentrated human habitation.