They don't distribute it evenly, the give it the government which uses the additional money instead raising taxes or cut spending. Of course the people who would pay the highest taxes profit most from this, because those people would otherwise pay the majority of the additional taxes, too.

But those people generally don't own much money, they have much wealth. That's an important difference here: Much wealth doesn't mean much money, it just means property which may be sold later for money. But unless someone really does this, there is noone with a huge stock of money. And why should someone trade his wealth for fiat money unless for short-term trading purposes?