Never did it myself, nor do I actually know anyone who did so I hope I'm not wasting bandwidth, but I think in the past couriers were expected to give up their whole baggage allowance for the client; traveling internationally on a budget with nothing but carry-on would be an interesting challenge these days.
I used to be pretty tapped into the budget travel scene and haven't heard much about air courier work in awhile; parcel freight (FedEx, DHL, etc.) just keeps getting better and security restrictions for passenger flights keep getting worse. Can't see it being a viable business these days.
Could be wrong. Let us know what you find out.