When you sign a publishing contract, you assign a portion of your song ownership to a publisher, traditionally 50%. Until you sign a publishing or co-publishing deal you and your co-writers own the copyrighted song. In the past ten years all sorts of new and differing publishing deals have come into vogue, probably, the most often used is the co-publishing - where you basically are entitled to the writer's royalties plus 50% of the publishing royalties. There also are copyright administration deals where you retain ownership of your copyrights, but assign a third-party such as a music publisher or songwriter's organization, i.e. SGA (Songwriters Guild of America) to administer your copyrights for you. If you decide to co-publish or establish your own publishing company you may want to consider incorporating or forming a partnership. If and when it gets to the point you're talking any kind of publishing deals, it's time to find a good, reliable, music business attorney. Check out your local Bar associations or, again, contact a songwriter's group for referrals to an attorney in your area
Each kind of deal has its own unique advantages and disadvantages which you need to be aware of. You might start off by contacting a group like SGA for information or check out a book like Randy Poe's "Guide to Music Publishing for Songwriters". I'm sure other board members have contacts and texts that they can recommend also.