I don't understand the purpose to this discussion. Let me give you Utah as an example.
When Utah first started issuing Concealed Firearms Permits you had to demonstrate proficiency with the type gun you intended to carry, revolver or automatic. If you wanted to be able to carry both you had to qualify with both. The requirement over time was reduced to simply qualifying with something, then to no shooting requirement whatsoever. In my opinion this last is the most reasonable situation!
Think about this, what is more important, to be able to shoot well, or to know when and under what circumstances shooting is permissable? I don't think there is any question which is most important! Several states have obviously come to the same conclusion. The ability of most people, well trained included, goes to hell when confronted with an actual shooting situation anyway!
To answer the original question, I don't believe there are any states that have absolutely no training requirements! So far as I know every state requires some training in legal issues! I hope that is what you were really asking! Of course there are a few strictly "Constitutional Carry" states that obviously have no training requirement. It would make sense if they had a requirement for training in legal issues though!!!
Why have some states rescinded reciprocity with Utah? Simply because they still have a shooting requirement and Utah doesn't. Why should this make any difference?