When I interviewed the CEO/founder of KEEN Software House (makers of Space Engineers) for my magazine, this was one of the questions I asked him. He told me that what really matters is that you get really good at making games. He said that in his experience, it makes no difference whether you get that skill from a school or by teaching yourself, but that you must get really, really good at it before you consider making it a profession. He said he got good at it by practicing and doing nothing but designing and coding things for years. He got most of his experience in a software company (not games), always waiting for the day when he felt he had the skills (and enough money) to get started. He suggested that you should focus either on code or on some specific design skill (3d modeling, graphic design, etc.) - and that the programming will make it easier to find work than the design will. He advised against trying to do both sides of it on your own, though he admitted you can get a lot of experience very quickly by trying to make games entirely on your own (making both the code and the art).
Many years ago I also had the chance to chat with the team who made another game (I think it was Asheron's Call - it was something I had never played but I went to a convention with someone). This was before I went to college, and I was considering going into games somehow. He told me that when someone applies to work with him, all he wants to see is a game demo. He doesn't care about a degree. If you can show him a game that you've made yourself and it impresses him, you've got a chance. To him, a degree means you finished a program in school, not that you're good at making games.
So I pass that on to you. : ) According to the only game designers I've talked to about this, school is generally not necessary - BUT it can be a simple and focused way to get good at programming or digital art, so it's not necessarily a bad decision if you want to get a degree.