Wow. Actually that fixed the problem I was having with the refraction, because the spheres were the same size.
And using one sphere with 2 materials was a great suggestion as well. It’s working much better, and much easier to work with.
I’m very close to getting it now.
The only problem left is that I still cannot get my transparency dark and my colors vibrant at the same time.
Here’s where I’m at:
This is globe with transparency and with no color, everything is good here:

When I add color, notice how the countries on the back of the globe are no longer as dark:

Just to clarify, I have one sphere, I have a material with alpha applied to it for the countries, and another material applied for the globe that has color, transparency, and reflection.
Here’s what I’m trying to recreate:

I can’t seem to get the same glossiness/reflectivity and bands of color like this.
Thank you so much for you help Adam. You’re a life saver.