Short answer: Formal education in music or audio engineering is necessary only if you think you
need guided education. You will know
more than you would without one, but it doesn't mean you'll be better than someone who doesn't have one. It's all based on you as an individual!
Long answer: The benefit to learning music production (i.e music theory & audio engineering) through a school like ADSR, Point Blank, ICON, or a more traditional art school or college is one of structure. Like Lydian said, all the information you'll actually need to be a successful music producer is available online - what brings you over the edge is how much work you put into becoming that successful music producer. There are risks to self-education, of course: There's a ton of misinformation (rarely intentional, of course), it's hard to know what to learn and when, it's very easy to get overwhelmed, and for most of it you are completely on your own. But there are plenty of people with no formal education in either music theory or sound design that learned enough to make a living off their tunes, so it's not impossible!
When you go to a school of any sort, you'll receive a degree planner that will show you exactly what you're going to learn and the possible orders in which you can learn them. You'll be given a syllabus with all the material you're going to learn in each class as you take them listing what that class is gonna go over and in what order, completely free of extra information that might confuse you. You'll have not only a teacher who is now being paid to pay attention to you and help you with what you're confused about between classes, but a bunch of peers all learning the same material along the same timeframe who you can feel comfortable asking questions to. If you attend a physical school, you get access to equipment and resources that you wouldn't have on your own (for example, places like ICON and Pyramind have labs where students can come in and use their studio equipment). Of course, all the formal education in the world means nothing if you can't put it to practical use, so it's not like going to one of these places is a guarantee of success.
There are three things you should consider before applying to any school offering a music production program or going to college for music/audio engineering:
- Is the school reputable? Does the school have accreditation and/or success stories? What kind of classes am I going to be taking? What kind of teachers do they have? What resources do I get for tuition? Does it look like they're trying to take me for a ride?
- How important is it for me to have the rigidly controlled learning experience in order to learn? Would I flourish better there, or would I work better learning at my own pace and studying things in the order I decide? Do I need a teacher, or should I just try to find a mentor?
- Can I afford the education the school is offering, or take out loans without putting myself in severe financial risk? Would I be better served saving my money and learning the material on my own? Do I want to pay money for what is essentially experience steroids, or do I want to spend that money on other things and rely on my own willpower to learn?