One of the advantages to living with another human is the division of household labour, but there may be a benefit to one person doing the cooking and after-meal cleaning.
At first blush, this sounds absolutely insane, but Emily from Cupcakes & Cashmere makes a pretty strong argument for the arrangement:
Doing things this way ensures that whoever cooks is held responsible for not making a huge mess, which actually minimises cleanup time and ensures that it gets done. If you know that someone else is handling the dishes, you're less motivated to clean as you go and use as few dishes as possible.
If you're the cook of your family unit, and are worried about an unfair chore distribution, I'm sure you can find something for your partner to do while you wash up. I suggest folding laundry, or perhaps putting away the dishes once they're clean. Putting away the dishes is the most annoying part anyway.
Our System for Keeping a Clean House [Cupcakes & Cashmere]