We have a dishwasher at the moment, but the things that need washing up by hand get done by whoever has time to do it. During the day that's me because I'm at home. At night it might be my husband if I'm busy putting one of the kids to bed or something.
My husband usually washes up as I am generally home first and cook dinner: I try to tidy the kitchen when serving up dinner: I unload the dishwasher while dinner is cooking: the arrangement is whoever is home first starts dinner, unloads the dishwasher etc. Manual washing up is confined to saucepans etc (stuff we do not put in the dishwasher). There is no hard and fast rule - whoever does what needs to be done.
PS My husband does the housework for our 2 bedroom/2 bathroom apartment: I generally do washing. ironing, cooking....but we do overlap often!
Dish washing is mainly done by our 'Dish Washer'. Like any other household chore all the work is done by us both. But i.e in caseof Dish washing it's simple if I load the dishes my wife has empty it & put the stuff away. well! if she does the loading then I do the job of unloading it & putting the stuff away
Simple n' no hassle.
I'm happy to say that that duty, one I despise, is one I share with my wife. I would hate to have to do all the dishes but no more would I want to push a duty I hate onto someone I love. So we split the chore evenly.
Our dishwasher, both of them, has always has been the best, the most reliable, the most economical, the most forgiving for awkward items and the most noisey on date nights.....our kids.
The only dishwasher I would ever pay for.