Throwing construction crap on the sidewalk...

They will not pick up construction related debris - it will be left and your building fined. Call a professional carting/disposal company. There are many of them, they are reasonably priced, and it is the right way to dispose of construction debris.

Actually, they're required to pick up small amounts of construction debris generated by the homeowner hisself (not by a tradesman). From http://www.nyc.gov/html/dsny/html/collection/bulk.shtml :

Construction Debris - While the Department does remove non-commercial waste from homeowners engaged in small "do-it-yourself" projects, homeowners who are undertaking large projects should contract for their own dumpster by contacting a private rubbish removal service. Bulk and construction debris generated by hired contractors or fee-for-service personnel on home repair or renovation projects is considered commercial waste and it is therefore the responsibility of the contractor to arrange for appropriate private disposal.

Good luck figuring out what's "large" and what's not, how many lbs. max per contractor's bag, how many per day, etc. Be sensitive to causing hernias and back injuries, but in no event should you tip a sanitation worker for doing his job ... it's not all styrofoam peanuts.

My experience as a DIY renovator was that they didn't pick up anything in construction disposal bags. You are right though, we did get away with small items in regular garbage bags. Be especially careful to make sure all nails are removed or hammered down before disposing - no reason to expose your sanitation worker to tetanus.