I used to do some cleaning for a few people for several years. A very experienced professional cleaner I knew swore by cleaning with cotton face cloths. I cleaned all the houses with them and have never used anything else in my own home since. No microfibre, no yellow dusters, nothing. Just face cloths.
No matter where the cloth is used (toilet or kitchen), they get put into a bag and once a week I put them in the washing machine on a 90'C cotton cycle, along with my tea-towels, floor mops, and anything else like it. The hot water and washing powder should be more than enough to clean the items. I tumble dry them after washing. I also used to wash the cloths when I was cleaning houses as it was too much to expect the client to do it, but this was covered in the price I charged.
As the client, it's up to you to decide what you want your cleaner to use in your home. If you are giving her disposable cloths for the bathroom and asking her to reuse dusters without washing, that's up to you, but as someone who only ever damp-dusts surfaces, I can't imagine how you could use a dirty damp cloth again without washing - goodness knows I sometimes use two or more in one room if I'm having a good clean, as they get too dirty to keep on using.
You may think that washing a toilet cloth in your washing machine is gross, but I'm of the mindset that cleaning your home with dirty cloths is much, much worse.
But as before, if your cleaner is providing clean cloths as part of the service, then it's your house and your call.