I usually throw it into the downstairs bathroom hamper, draping it over the edge if it is wet, then throwing it in when dry. If there is a basket in the kitchen waiting to go downstairs to the laundry room, I'll toss it in there. DH usually just throws it on the floor by the stairs (and l will put it in the hamper or downstairs). I've asked him not to, since the dog will chew on the towels if they have something tasty on them, but whatever.
My plan is to make a wet bag (like for cloth diapers) that has straps that snap onto my oven handle. I've seen similar WAHM items on etsy and hyenacart but I love to sew my own stuff. I was thinking of doing a mesh panel on the back and a cute print on the front so stuff will breathe but I won't have to look at the dirty towels.
I do wash my kitchen cloth as its own load because we have a lot of it. I like a plastic and paper free house as much as possible so we don't buy paper towels, paper napkins, plastic bags etc. I have several dozen cotton dish towels (some are wedding gifts from 18 years ago!) and I tend to go through them pretty fast, several a day. Plus the placemats and napkins get changed every couple days, then there are the food wrappers and bags, microfiber cloths for cleaning, etc. By the end of the week I have enough kitchen cloth for a load. I use Charlie's for all wash loads and for the kitchen load I will sometimes add a bit of borax or Bac-Out if there are particularly icky things. Kitchen cloth and bathroom cloth are the only loads I wash on hot.