I work with a person who receives a 1099 from a corporation and I don't. We share the costs of supplies and other expenses and he pays me for the work I do. I would like to be able to claim my income. How can I do that and pay the taxes on the money that I earn? Does he have to fill out a 1099 for me for working with him?

Yes, he must originate the Form 1099-MISC for you as a subconractor to him.