From a self defense or hunting perspective, having both eyes open is a real benefit. The eye not used for sighting is still able to catch movement even if you aren't concentrating on that eye's view. When you close the eye not being used for sighting, you are making a giant blind spot/blind side in your visual awareness which is already being reduced when sighting with the other eye.
I picked up the practice in the family store, using a loupe and dealing with customers. It is amazing what customers will try to do when they think you aren't looking and that your whole world field of view is concentrated down to a square half inch. One eye on the jewelry and one eye on the customer.
The doubling of the targets as noted above is a problem some have. Supposedly, it can be overcome with some simple training exercises, but I have never done them and don't know how well they do or do not work.
"If you look through your scope and see your shoe, aim higher." -- said to me by my 11 year old daughter before going out for hogs 8/13/2011
My Hunting Videos https://www.youtube.com/user/HornHillRange