The laundry staff takes your stuff out of the paper bag and puts tags on each item (it's fun finding the tags later...like 2 years later), then transfers everything to a net bag for washing. They don't care (and how would they know for sure) which items belong to which person in the cabin.
Be aware that some have had clothes returned with shrinkage, holes, etc. Never happened to us, but it could. I wouldn't put anything that's brand new or expensive in the laundry bag, just in case. Some have had success getting reimbursement for ruined items.
Also, to verify what's being sent, I take pix of all the items laid out on the bed. That way if something is missing, I can report it and hopefully get it back. Same as above, never lost anything, but it has happened to others.