It really depends on the designer and the agreement you strike. I usually commission my artwork based on my ideas, and I pay for the rendering so the artwork is mine once payment is transferred.
In your case, you would need to come to an agreement with your graphic artist about who owns the design, and if not for what length of time you can use it: 500 prints, two years, etc. I would put the terms in a contract as this is a business relationship.
Regarding enforcing the illegal reproduction of your designs once online? You need money to enforce those rights, and for those offenders who are outside the United States -- if you are US based, how to you even get their Courts to enforce your rights to the art?