The waste is still collected by truck from the terminal, and presumably driven to somewhere else in New York.

London tries to reduce large vehicle traffic by taking the waste away by barge; you can see the place in central London here [1] (I like the irony of "Smuggler's Way".) The Roosevelt Island waste terminal is at the edge of the island, so this is a missed opportunity -- if there's anywhere to accept it by barge. (London's used to go to [2], a quick search suggests it now goes to [3].)

[1] https://www.google.co.uk/maps/@51.4628165,-0.1927893,296m/da...

[2] https://en.wikipedia.org/wiki/Mucking_Marshes_Landfill

[3] https://www.google.co.uk/maps/place/Cory+Riverside+Energy/@5... / https://en.wikipedia.org/wiki/Riverside_Resource_Recovery_ER...