Assuming you are seeking radio airplay in the U.S. every songwriter and music producer/composer should register themselves as a member with one of the three pro's in the U.S. -ASCAP, BMI or SESAC. Additionally, every songwriter and music producer/composer should register as a music publisher with at least one of the three pro's in the U.S. While you may only be a writer member of one of the three pro's at a time you may be a publisher member of all three simultaneously.
If the artist didn't write their own lyrics, traditionally they wouldn't be due any performing rights royalties from radio airplay from ASCAP, BMI or SESAC. I say traditionally because there may be some super star recording artist that may negotiate receiving performing rights royalties from radio airplay from ASCAP, BMI or SESAC of the song they recorded. While this is probably rare it's possible given that a new music producer or songwriter is seeking to get a placement and have that placement adequately marketed and promoted as a single by the record (new media) label.
If the recording artist writes the lyrics they are performing they would receive the writers share of the performance royalties from radio airplay from ASCAP, BMI or SESAC.
It may be extremely difficult to get anyone who knows about the music business and music publishing who intends on investing significant marketing monies for radio promotions to buy a beat non-exclusively. Anyone who agrees to such an arrangement may be naive to the music business and will ultimately turn in to a headache down the line when they get the skinny. It is typically better to deal with people who know about the business, so that everyone is one the level, as this helps to ensure all parties remains happy when the money starts coming. When selling a track non-exclusively this would be leasing a beat or track and the terms of the agreement are to be decided by the parties involved.
Always have written music business contracts.