I am wondering, whether or not this results in a discounted hero power if the same player plays these cards. This page on sideshow spelleater claims that cost reductions gained by the opponent would not transfer. Hence my guess that the battle cry doesn’t affect costs.
To understand Raza better: It doesn't make your hero power free. It gives you an "aura" effect. This aura makes your hero power cost 1, whichever hero power that currently is, or changes to. This aura effect can not be changed (so far by the current rules of the game) and last untill the rest of the match.
So if your oppoent has achieved a 0 mana hero power through Raza and you play Sideshow Spelleater, you will copy their hero power. But it will cost the standard mana cost for you.
If you use Raza and THEN Spelleater, you'll steal your opponents hero power and it will cost 1. Think of Raza of only giving the selected player an aura that affects the player, not the hero power itself.
To be more specific, the card should say "THIS PLAYER WILL HAVE THEIR HERO POWER COST CHANGED TO 1".