Are airlines responsible for Airport security or Airports concerned or third-parties on terminal-side?
In the United States of America passenger and baggage screening is handled by the Transportation Security Administration (TSA). It is a government run agency and deals with the pat downs, X-Ray machines and frisks. This will get you through "security" and to the terminal area and ultimately on to the plane. 

If you are coming in from another country you will need to pass through customs which is managed by its own agents and falls under the jurisdiction of U.S. Customs and Border Patrol who are also a federal agency. 

At some of the really busy airports here in the US its not uncommon to see local police officers helping out with things like traffic flow and passenger management of those entering and exiting the facility and security in general.