In Godot, the result you are getting is obtained by having the
double sided option enabled on the material, so I think you have to disable it. If materials are directly in the mesh however, I'm not sure how you can do this, maybe something in Blender to prevent backfaces from being rendered, or add a material override in Godot.
However that doesn't fixes all cases, I tried with Suzanne (the monkey from Blender) and you can see outside faces through the model. In theory, there is nothing wrong with it, because you can see through the model (because it's transparent) and outside faces are supposed to be seen. What you want (judging by your screenshot), is to not be able to see anything of this model through itself... I don't see any option in Godot to do that, unfortunately :(
I think there are ways, of course, but not as simple as setting flags in a material.
It has to do with the depth buffer, like, drawing the mesh as if it was opaque, but the result must be made transparent. Maybe use a render-to-texture? (but may come with drawbacks).
If anyone knows, correct me if I'm wrong, because I'm not experienced yet in the 3D side of Godot 2.x ^^"
However if disabling
double sided is enough for you, I might move this comment as answer.