According to the city health code, technically, no, they aren't supposed to poop on the sidewalk. But they can poop on the street, as can your dog. You could, in theory, attempt to find an official of the Departments of Health, Sanitation, or Parks and Recreation to issue a ticket to the officer if you should find them letting their horses create a nuisance on the sidewalk again. I would not care to speculate on the results of such a maneuver.
http://www.nyc.gov/html/doh/downloads/pdf/public/161-03-notice.pdf§161.03 Control of dogs and other animals to prevent nuisance.
(a) A person who owns, possesses or controls a dog, cat or other animal shall not permit the animal
to commit a nuisance on a sidewalk of any public place, on a floor, wall, stairway or roof of any
public or private premises used in common by the public, or on a fence, wall or stairway of a
building abutting on a public place.
(b) Notices of violation for failure to comply with this section may be issued by any authorized
employee, officer or agent of the Department, or of the Departments of Sanitation and Parks and
Recreation, or successor agencies.
