This is such a tough question, and always seems to be a topic of heated debate when it comes up (not least in my office, where we have two trained designers, a boss whose design training is in different disciplines and who took a very roundabout route into interior design, and an assistant who wants to be able to be a designer without having to shell out the serious cashola demanded by school). A lot of people firmly believe in working their way up the old-fashioned American way, whereas others are fiercely protective of the value of their education and training. My answer? It depends.
For most people, I would say that to get an entry-level interior design job, you need at least an associate's degree; and if you think you'd like to work on commercial projects like restaurants, bars, hotels, and stores, then you'd do well to get a bachelor's, which offers more in-depth training on topics like space planning, building codes and safety guidelines for public spaces. Now, the assistant at our office is a perfect example of the fact that you can learn everything you need to know about the administrative aspects of residential projects on the job: she places orders, keeps project binders, oversees installations, and shops for fabrics and decorative accessories. But what she can't do is draft, sketch, space plan, or, if she leaves our firm and tries to get a junior designer position somewhere else, show a portfolio of work highlighting her design and presentation skills. Which is what everybody else she's going to be competing against for that job will have.
And even aside from providing basic skills that are required for the job, school hones and refines your eye in a way that working can't, and in a much shorter time frame. Your projects are for imaginary clients with infinite budgets and no objections to anything you suggest, and you're presenting your work to professors who push you to make it as bold and creative as possible. The constant, intense competition with your classmates, while it gets you semi-hysterical and strung out on Redbull in the days leading up to a presentation, also forces you to push yourself harder than you ever realized you could. All in all the process is basically identical to the forged-by-fire montage they show in those Marines commercials.
But notice I only said "for most people" a degree would be necessary. I think that since the field has become so popular, and design education so much more structured and competitive than it used to be, anyone looking to get hired by a company is going to be expected to have it. Which means the way around it is to open your own firm. If you are lucky enough to find people willing to pay you money to design spaces for them, it couldn't matter less what classes you did or didn't take, as long as your clients are happy. There are a lot of designers out there with no formal training, and some of them are really good. Many of them transitioned over from other visual/design disciplines. But the reality is, to get where they are, most of them had either great connections or great wealth, or both. That's not bitterness talking, that's my objective observation.
So do you really need to design training to be an interior designer? Most of the time, yes. For me, absolutely. The difference between where I was when I started school, barely able to draw, and where I was when I finished was nothing less than astonishing. But I was coming from a liberal arts background, where writing analytical papers about art history was the closest I got to design, so I had a lot of ground to make up. But if you've already got a solid art or design foundation, and can take a few drafting classes at your local college so you can learn autocad (which you DO need to know, no matter what, and you MUST know how to draft and sketch in scale), and you have clients ready and waiting to ply you with money, then fuck it. Go for it! It's the American way.