Putting together a few decks with these guys, question... if I have training on one (hopefully two!), once I defeat a character, am I rolling in both die on that now elite executioner?

I was gonna ask the same question. I am aware of the precedent set by the clone trooper but the wording on Palpatine from SoR also states this die which is a much different ability but also makes the wording and rulings on these cards confusing at times.

This die refers to a single die, so it can only trigger one die being rolled in. As noted, this has precedent with the Clone Trooper.

As for Palpatine, he is different because each base die is resolved independently so it creates two different trigger events which each effect a single die. So each time his ability triggers it is still effecting a single die just like the other two characters.