I've one wondered about the sanitation in New York City...Now I have a closer look.
Seems like the landfill of Fresh Kills, Staten Island is carrying the burden of all 5 boroughs. Some of Staten Island's residents may object, but so far, it seems the only remote enough place from living neibhorhood, plus the counter benefit is that the methane produced from underground garbage is quite useful.
Seems that this Fresh Kills business for garbage dump has halted since 2001 (http://www.ny1.com/content/top_stories/136003/parks-department-celebrates-fresh-kills-transformation)
And all boroughs are responsible for their own garbage: Shipping out-of-state via rail or sea.