I’m sure it depends to an extent on the buildings, I can’t say how they do it in super ritzy buildings. But when I lived in a lower middle class/working class neighborhood all the apartment buildings just had a row of buzzers at the building entrance with each apartment number on them, and each apartment had a speaker inside it that may or may not work to talk to whoever’s down there, and you go down to pick up your stuff. If you’re really unlucky they can’t hear you reply at all (either from street noise or one of the speakers not working) and you have to race down however many flights of stairs to meet them.

Edit: oh, I forgot you could also buzz them in, but they usually wouldn’t come up the stairs.

They do go up the elevator right to your apartment door. Why wouldn't they?