It depends on who they are.
If the songwriters are pop writers like Max Martin or Ryan Tedder, they carefully craft a hit single, find a boy band or celebrity diva to sing it, and rake in the millions. The music publishing company takes a cut, and the performer responsible for the recording, and the record label. But if the hit is a hit, album sales, and streaming, and radio plays, and use of the song in movies and television and ads are usually significant enough that the songwriter makes bank.
If the songwriters are indie musicians like Rennie and Brett Sparks of the Handsome Family are, things work a little bit differently: they write their own music, tour and record albums themselves to promote it. On top of that, they take what they can get in royalties on their compositions.
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