First, some context. I have degrees in math and experience in computing. I am now part-owner of an established company, and occasionally we recruit. The following is my experience based on some successful hires, some unsuccessful hires, and many, many CVs and interviews.
When I look as a CV I want to know what value you'll bring to my company. I want to know why you'll be worth the money I'll pay you, and how well you'll fit. Yes, there was a job description, but your role will change, develop, and in the end you may be doing something completely different from what we originally thought we needed.
So I want to see, quickly, what value you claim you'll bring to my company. That means you'll need to have thought about that, and tailored the CV to match not just the job description, but the homework you've done on what we do. In particular, that makes it hard to get a job via a recruitment firm.
After that, I want to see the evidence that your claims are true. Do you claim to write clear, usable, effective code? Show me some code. Do you claim to work well with others? Tell me of specific occasions that I can verify.
Sometimes you can't provide evidence, and I understand that. Make it clear that you understand that I will somehow need to verify your claims. Be sympathetic - put yourself in my place, and help me see you as a potentially valuable asset.
And all this is true even if you don't have a degree. State that you know things, and then provide evidence. I've employed people with and without degrees, and it's been interesting to see how different people have worked out.
tl;dr - No, you don't need a degree. Yes, you can get a job, but you need to provide evidence of value other than by quoting that you have a degree. Sometimes that's better.
Edited for typo and word-crafting.