Performance Royalties consists of 2 types of shares: Writers & Publishers.
If you are a solo artists with ASCAP, you would need to join as a songwriter and publisher to collect both shares. Many artists miss out on royalties by not joining as a publisher with ASCAP, and that money falls into what we like to call a "black box".
As far as BMI, they operate slightly different than ASCAP. If you're registered with BMI as a songwriter and not a publisher, you would receive the writer's share and the publisher's share would automatically default to you. According to BMI, if you are just getting started, you don't need to set up a publishing company. Although it is recommended.
When there are multiple writers, the publishers' share is tied to the writers. So, if a writer owns 50% of the writer's share, their publisher would own 50% of the publisher's share.
If a co-writer wishes to retain 100% of the publishing, all parties must enter into an agreement stating that the publishing company would represent all writers' publishing share.
To set up a publishing company, you would need to form a business entity and register the business as a publisher with your PRO.
Feel free to check out our e-book, The DIY Musician's Starter Guide to Being Your Own Label and Publisher on our website, that details how you can become your own publisher. You can find it here: https://www.tuneregistry.com/lp/the-diy-musicians-starter-guide-to-being-your-own-label-and-publisher