It’s the responsibility of both, but this is the least of my worries with laundry.
Both my husband and I dislike doing laundry, but I dislike it less than he does. My biggest battle with it to date is finding all the dirty laundry, as he frequently will deposit dirty laundry wherever he undresses. I’ll scout around the house, looking for things to wash, and ultimately, after I’ve done all the loads, I’ll find errant dirty socks and shirts in places I never thought I’d have to look for them. Since I do the laundry, I double-check most of the clothes for pens, etc., but lately I’ve gotten a little lazy with this task and will just wash his stuff that is likely to have those things separately. Why? One time when I was washing our stuff communally, I didn’t check well enough and an open sharpie marker got into the wash. It ruined two pairs of pants and a few nice shirts of mine that I had for work. I had to run out and get dye to recolor those pants, and I had to throw out the shirts, as there was not a good solution to hide the dozens of permanent marker streaks and dots all over them. :smack::smack::smack:
Also, if I have him do laundry, the likelihood is higher that things will get shrunk or discolored. It’s not entirely his fault, as he has no idea what to do with dress clothes and laundering.