In a socialist system you still need a government, which is a group of people who are empowered to enforce the rules of socialism. As a result, they end up having access to most of the collective wealth as well.

If they are very good socialists they will redistribute it all. If they are not-very-good socialists they will redistribute some of it and reserve some to support a nice lifestyle for themselves and their families. They won’t personally “own” mansions, airplanes, factories, etc. like capitalists do, but they still control them legally so the practical effect is very similar.