Communism means a moneyless society. There is no “earning for your labor” anymore and in turn no real loss for the artist that prevents them from having access to basic necessities, nor any real incentive for others to illegally copy someone’s work in the first place. So no, communism doesn’t really need any kind of copyright. Making it mandatory to credit the original artist could be mandatory, but that’s about it.

Though I do agree with your second paragraph for the time being, where money earned through labor is required to survive.