I couldn't care less if someone has a degree or not when it comes to game development. I have a degree in journalism so go figure.
Depending on what you're interested in, in game development, there are a number of things we look for.
I'm an FX artist, so the only thing I'm looking at is your demo reel. If I like the reel I'll take a look at your resume and if you have a degree listed or schooling, that's cool, but I really don't care honestly. I'm more interested in the reel and then after that what games you have worked on, if any.
For level design it's similar. They want to see what levels you've designed and see if you know your way around creating a good level.
For programming/engineering -- this might be the one where you'll need schooling, but I have no experience hiring or knowing what they look for in an engineer.
For world art, again they want to see the different models you've created, see if you can create an asset that has good poly count and good UV economy. And they'll want to see how you put those models together on a level maybe... see if you can "art" a basic level.
All of this can be learned online for free. As for engineering, again I'm not sure if you need a degree in that or not, I'll leave that part to someone who knows.
I would suggest you download Unreal. It's free and it's the premiere game engine/editor. Blender, also free, is fast becoming adopted in the video game world, but I would say to check out Maya and 3ds Max as well. I think they both have free versions as well.
Start with Unreal, have fun. Then check out Maya, play around with it a little. Then check out Max and Blender and do the same. Whichever one you like best, focus on that package (Maya, Max, Blender).
Then figure out what you want to do.
If you just want to be a designer, you don't have to even worry about Maya, Max or Blender. Build all your levels in Unreal, make a really cool level and go to the different forums (Unreal has it's own dedicated forum but there are many many more) and ask for feedback.
If you want to do FX there is a forum realtimevfx.com Check out that forum and see how some of the people are doing the cool FX they do.