Would you need any schooling to become a successful jewelry designer?

No. Not if you got a good sense of design and business. However being certified gemologist might help. To learn it, consider being an apprentice to a designer or working for a jeweler who makes an repairs jewelry.
Designing jewelry is one facet (pardon the pun), then you have to sell it to people, and promote your company. That's entrepreneurship and most experts today say that is not taught in college. There maybe college courses for this, but I would think other avenues as well.

Associate degrees in jewelry design typically consist of 60 to 90-credit hours and can be completed over the course of two years. Program curriculum will consist of general education courses as well creative design courses. Course titles include Metalsmithing, Color & Design Theory, Jewelry Design Ideations, Art History, Gemology & Gem Identification, History of Jewelry, and Professional Practices. These programs will require a lot of hands-on work in which students will have the opportunity to design and create their own jewelry pieces in design studios and labs over the course of their studies. Some programs will also require students to submit a completed portfolio or collection at the end of the program just prior to graduation. Upon graduation designers will be ready to take what they've learned into the professional field. Some may choose apply to jewelry companies, while other will go on to establish their own businesses. The majority of these programs can be found at art institutes and technical schools that specialize in the field of design. Applicants interested in earning an associate degree in jewelry design should have a high school diploma or GED. Due to the design nature of these programs most applicants will also need to submit an entrance portfolio to showcase their creative skills and abilities. Students at this level can expect to pay anywhere between $4,590 and $35,290 in annual tuition, depending on the school.

http://www.collegesanddegrees.com/programs/jewelry-design

Hey Daphne,

First, let me give the typical answer and suggest that even if you do not go to school for jewelry making, you should consider going for some kind of degree, so that you can achieve a career if the jewelry making does not work out. If you got a business degree, it would make you into a better seller which can further your business.
That being said, check local and community colleges for classes that wil enhance your skill, like glass blowing, or a class that teaches paper work so you can make your own packaging. Using these to set yourself apart from competition will help. Once you have these skills, you can partner with other businesses to sell your work.
Enjoy!

Daphne - You do not need to go to school to create jewelry. However, what you are going to do with the jewelry and whether or not you will work for yourself or someone else might make school more important. School cal help prepare you for some of the business side of things. It can give you things to think about that you might not have considered. It will help you network and introduce you to people and concepts that could assist you with promoting your jewelry. I cannot answer if school is or is not for you, but I wold say it depends on what angle you will take your jewelry in the future. You might talk to other jewelry makers and ask them what they did, what they think and what school could do to help you if you pursue a similar career.