I don't buy purpose-made shop rags because I believe it to be wasteful.
Instead, I buy scraps and remnants. You can order them for about $1 per pound from places like this:
NEW PIG
Also, you can usually go to garment factories and buy the remnants for about $0.40 per pound.
I wouldn't mind sending them to a recycler when they are used, however they may be considered a "hazmat" once they are oil soaked. Unless the service picks them up, I wouldn't spend the money to ship them. Just put them in a fire-safety can until bin day.
I don't think it is wise to launder them yourself due to the buildup in your drain system. And if you are on septic, you can poison the tank and/or poison the ground water.