Specifically to your circa 1700 question, one thing worth remembering is that when we are talking about "cities" in what would become the modern-day US, we are talking about settlements that were by our standards tiny in population. Boston in that period had about 7,000 inhabitants, New York about 5,000, and Philadelphia about 4,500. Which is to say that these were much closer to towns than cities - Boston was literally a town, with a Board of Selectmen and Board of Assistants and not a chartered, incorporated city with a mayor until 1822. These cities/towns/what have you were also geographically small compared to their modern equivalents - Boston in 1700 being not even all of the peninsula which today is downtown Boston, and other modern-day neighborhoods of Boston being at the time miles-distant farming communities, if existing at all.

In Boston during this period - yes, inhabitants had privies that either discharged into open gutters along the street, or into cesspools or vaults that would leach into the soil, or be periodically cleaned out. It wasn't until the early 19th century that Boston developed a "dry sewerage" system, whereby residents would transport privy contents themselves to a central location.

Basements and houses faced periodic flooding by household waste and storm water. Drains (laid under roads) constructed to deal with these situations were built on an individual basis, or by a group of neighbors who would charge an access fee, and were not town projects. These drains were more properly to deal with storm water, however, and not sewage per se - generally sewage would go into the afore-mentioned cesspools, and either be buried or cleaned out when the pit was full (it was possible to hire a contractor to clean out the cesspool: this was good fertilizer material, after all).

The city of Boston took over the drainage system in the 1830s, and sewage was allowed to be dumped through the system in 1833, which caused worse sanitary conditions: as mill ponds and tidal flats were filled in and developed, waste had less places to go, and more often than not festered in the remaining tidal flats at low tide in the harbor.

As Boston grew into a larger city, these systems became increasingly untenable, and contributed to major cholera outbreaks in 1849 and 1866. In 1875 the city undertook a study to build a modern sewage and pumping system, but this wasn't completed and operational until 1884.

Anyway, it's worth noting that human waste was not necessarily the biggest sanitary concern to inhabitants at the time. These settlements would have been filled with animals, live and dead and their waste. Besides horses, Boston would see live pigs, calves, lambs and goats brought to market regularly, with 1692 regulations attempting to control how such animals were brought to market (live animals could not be slung over the sides of horses, for example).

Butchered animals provided even more sources of complaints. As early as 1642 a Boston butcher was ordered to remove his "stinking garbage" from his yard, and in 1656 all butchers in town were ordered to dispose of their offal - by throwing it into the tidal mill creeks that (in theory) would carry the waste out to sea. This wasn't the end of sanitary concerns of butchers: a 1735 law prohibited their selling of "blown up or winded" meat (ie, meat with blowfly maggots or taint) - the fact that such laws needed to be passed repeatedly showed that butchers were not abiding by them. Midcentury there was a push to relocate butchers out of town, and by the 1780s there were none within the town's limits.

In summary - the settlements and their populations of the time were much, much smaller than we are used to in US today. It's basically true that they did not have developed urban public drainage and sewage systems well into the 19th century, and yes, waste was disposed of (or not) in an individual fashion. But as far as smells went, these were just one of a number of contributors, as many live (and dead) animals would also provide a host of pungent aromas in town as well.

Sources:

University of Massachusetts Boston has a blog by public history graduate students under Dr. Jane Becker who discuss this topic, including a post on the history of sewers in Boston and a post on early sanitation.

On the animal front, there is Karen Friedmann. (1973). "Victualling Colonial Boston". Agricultural History, 47(3), 189-205. At JSTOR