Two singles could be washed separately, or on different days or weeks.
Mine is synthetic filled but has a cotton outer. I had to search to find it, most of them are acrylic, but natural fibre ones do exist. I already used a single ordinary one from Ikea, which has a white cotton outer, which I used without putting on a separate duvet cover, and washed every two weeks or so - very successfully, I could put it back on the bed the day it was washed. However it was only four and a half tog and not warm enough in colder weather, so I looked for (and bought) a coverless cotton one. I am very pleased with it so far.