Public Execution targets only the creature it destroys. If that creature is an illegal target when Public Execution tries to resolve, it won't resolve and none of its effects will happen. No creature will get -2/-0.

Only creatures controlled by the player when Public Execution resolves will get -2/-0. Creatures that enter under that player's control or that the player gains control of later in the turn are unaffected.

If the target creature regenerates or has indestructible, each other creature controlled by that creature's controller will still get -2/-0.