How do non-US pilots get the hours necessary for an ATP?
Under the rules of ICAO Annex 1, Personnel Licensing, you don't need an ATP License to fly as second in command (SIC) on a commercial aircraft required to be operated with a co-pilot. A commercial pilot license (CPL) is sufficient to be SIC on an aircraft that requires to be operated by more than a single pilot. 

And since 2006 ICAO have defined a Multi Crew Pilot License that also allows a pilot to be second in command on a commercial air transportation multi-crew aeroplane. 

The  ICAO website about Multi-Crew Pilot Licence (MPL):


  The MPL allows a pilot to exercise the privileges of a co-pilot in a
  commercial air transportation on multi-crew aeroplanes. It provides
  the aviation community with an opportunity to train pilots directly
  for co-pilot duties. It is a new licence that has been introduced in
  addition to the existing pilot licences defined in Annex 1 — Personnel
  Licensing.  
  
  The licence focuses on ab initio airline pilot training.
  MPL training and assessment will be competency-based and involve a
  multi-crew environment and threat and error management from the onset.
  It provides for greater use of flight simulation training devices and
  include mandatory upset training. At this stage, only aeroplanes are
  considered for this new licence. The details of the requirements for
  the licence are contained in Annex 1 — Personnel Licensing and in the
  Procedures for Air Navigation Services — Training (PANS-TRG). These
  documents outline the minimum international Standard for the
  implementation of the MPL by any State; they can be purchased directly
  from ICAO through the Document Sales Unit.


In Annex 1, the requirements are laid down for the MPL. One of the requirements says: 


  Experience
  
  2.5.3.1 The applicant shall have completed in an approved training course not less than 240 hours as pilot flying and pilot
  not flying of actual and simulated flight. 


The MPL, combined with a type rating, will get you to the right hand seat of the aircraft of choice. 

So it's likely there are 250-to-500-hours commercial pilots flying Second In Command (SIC) into JFK. With experience in Beech Bonanza's, Cheyenne/Seminole, maybe CitationJet, a lot of simulator time and a couple of hours in a B777.

To answer your title question, in many countries pilots get their ATPL after building hours as SIC flying with a CPL or MPL within an airline training scheme.