My experience is in the US market only.
It's not clear to me if you want to do software or hardware. You mention shooting for a EE position, but you also say you enjoy embedded programming work.
Which do you want to do? Yes, you have to choose. These two disciplines have some overlap, but not as much as you might think. Personally I do both, but professionally I've never really seen it work. Being a EE for a company is a full time job and so is being an embedded engineer.
If you want to be a EE, then you really need a EE degree. Yes, there might be other paths to that goal, but all things considered, this is your best bet, IMO.
Self teaching is fine (great, actually), but you need the paper.
If you want to be a embedded engineer, you might not need a specific degree. 15 years ago, you certainly didn't need one, but things are changing. Many embedded folks have EE or CS degrees now. That's the majority of your competition.
I think I remember an episode Dave gave on interviewing that talked about bringing your projects to the interview. I think that's an awesome idea. It might help you close the gap between yourself and people with a degree but no real passion. However, there will be a fair number of people that have both. That's really the bottom line. You want to have both.
All that said, I think it's possible to teach yourself the equivalent of what you'd learn as a EE or CS major. Definitely not easy, but possible. However, very few companies are going to take a chance on you unless you can really blow their socks of with your creations. And if you can do that, you really shouldn't be working for them--you should be working for you.