If I switch airlines during a flight, do I have to re-check my bags with the new airline?

I'm flying to Buenos Aires, Argentina. My first flight is on American Airlines it goes from New York to Miami. In Miami my next flight is on COPA and it is a direct flight from Miami to EZE in Buenos Aires. When I arrive in Miami, will I have to get my bags from the American Airlines flight and re-check them with COPA? I am buying the two tickets separately because it is cheaper than buying it all at once.