I have this same non-background and work on the proverbial team of mostly PhD's. Short answer is yes, you can do it. Long answer is that you have to be really, really good to compensate, and getting to that point is absolutely exhausting. It's not about just going through a couple ML courses on coursera. You need to understand statistics, CS, and ML at a really deep level, and that means being good at applied math too. I was lucky to come out of physics and have a solid applied math background anyways, giving me a few years head start on that self study.
If you need structure to go through a few years of coursework on your own, you should go for the degree. If you just want to learn how to put pieces together and not learn how/why they work under the hood, you should opt for something else.
As with most questions about going nontraditional routes, you have to be really good to compensate, and getting really good is constant exhausting work.