Also depends on what type of filmmaker you want to be. Some film schools train you to be a industry standard filmmaker/technician. They teach you how to work within a studio setting. Plus some have direct connections with industry (with studios, production houses, internships). That's important if you want a studio career (as sound mixer, DP, editor, post supervisor etc...).
Also schools give you connections with industry through the teachers, alumni, and even your co-students. These connections can (if the school is good) carry your career for the rest of your life. You'll get more work through your connections than through your resume during your entire career.
Other schools will take your money, give you a good basic training, and then leave you at the door when you've graduated. That's fine if you're a self-motivated person and realize that going in. But you have to be a very motivated person. Otherwise you may feel, well what now? afterwards. And then struggle to get any industry work after.
What I always say too anyone who wants to be a filmmaker is, "what's your day job?" If you can get a good paying day job (preferably in the industry) then that'll give you both the money and freedom to make your movies. Even then you have to be motivated to work on your film after a hard days work on someone else's. Remember, if you want to make personal, director-driven feature films, you won't make any (or very much) money from them (I'm generalizing here, but that's often the case, 99.9999999999999% of the time). You get the idea.
On the other hand if you want to direct commercials, industrials, episodic TV, or other people's work then find a school with industry connections. Even if you don't necessarily want to do any of those, still find a school with industry connections. Some of the best filmmakers I know, in fact all the best filmmakers I know, have day jobs in the industry - working for studios and production houses as motion graphic designers, editors, DPs, animators etc.
It's not impossible to be a filmmaker without going to film school at all. It's just harder. A lot harder. For one reason. Film school gives you connections. You'll network with other students and be able to find people you'll work with for possibly the rest of your life. It's easier to find potential Producers to work with (on your magnum opus) at film school that trains Producers than outside in the world. Same goes with putting a crew together.
But if you have the drive and are incredibly motivated you could spend your money on making a feature film and getting a crew together without going to film school. But finding/making the right feature and making it look as good as it should be is another story. This approach has as many pitfalls as anything else. I've seen too many feature films that could have been so much better if the makers had taken some advice before (and for example, shot with a different camera/lens, cast better, picked a different genre etc).
Personally, if I could afford it, I would go to a film school. But I'd make sure it was one that could get me (or help me get) an industry job when I left. Talk with the school on their success rates of finding students work in the industry. Speak with alumni. See what guild, association and union connections the school has. Basically do you due diligence.
But having a day job is what every filmmaker needs. It's gives you the freedom to take your time preparing to make a film, then time to make the film and the freedom to not have a heart attack when you finally realize you won't make a penny once it's released. Sure, your film can make money, I see that with films I distribute. But it's never as much as you'd hope for. Once you have a job, then you can make a film.
Hope that helps.