Are commercial pilots allowed to fly internationally? Are Commercial Pilots required to take an Airline Transport Pilot (ATP) program/training to fly internationally? And is a commercial pilot the same as an airline pilot?
A US rated pilot (private, commercial or ATP) can operate an N- registered airplane anywhere on earth (subject to the persons ability or inability to enter certain countries airspace).
At the regional airlines it was common that the first officer only possessed a commercial certificate (the full ATP is earned during the pilots first upgrade training to become a captain). These first officers will have no restriction on international travel based on their certificate and can operate international flights. These days FOs will have restricted ATPs rather than commercial certificates, but this is unrelated to the ability to operate internationally.
Outside of airlines you'll find positions where the only a commercial certificate is necessary and these pilots can operate internationally.
If the airplane in question requires a type rating then if the copilot does not have a full type rating, they will need an "SIC type" to operate internationally. This stems from an ICAO requirement that both pilots be type rated. The SIC type can be added to a private or commercial certificate, not just an ATP so this won't be a restriction based on certificate level.