Experience
May, are you the one who left a comment on the entry yesterday? :] A writing degree isn't mandatory, but it does help. At the end of the day, though, it doesn't matter where you get the skills so long as you have them. You can have a master's degree in English and still be a crappy proofreader, and you can have a GED and be an amazing proofreader. It depends on your motivation and your willingness to learn.
kk
Very true. It's getting the experience without a degree than can be troublesome. At least getting the kind of experience employers will count higher than a degree.
This is just my experience, but I think the problem often comes from setting a specific goal right off the bat. Just get around the business. Don't set your sights on being an editor, a proofreader, or anything else right at the start. Get around the business, do whatever you can find to do, take whatever position you can find, and use that position, even if it's working in a mail room somewhere, as a stepping stone to a higher position.
But I will say this; proofreading is seldom a position. Most often, a copy editor does the proofing. At most places, one of the copy editor's prime responsibilities is proofing, so there's no need to hire a second person as a proofreader. It would just be a waste of money.
There are courses in copy editing, and such courses can replace a degree. You must, of course, have impeccable knowledge of grammar and punctuation to be a copy editor.
My favorite copy editor site is
http://www.theslot.com/ Bill Walsh is one of the best, and the Sharp Points section of the website gives a good idea of the kind of knowledge a copy editor, and a proofreader, must have.