Not sure why so many are saying this is no longer going on? Why would you think so? I am working as flight courier for three companies in the last 5 years. Work is irregularly, as its very much on demand - about 2-5 times a year, irregularly. So its not like a job or something. More like an opportunity to head to some destination at short (more like very short) notice, get some air miles and get paid a little for your trouble.
All three companies I am registered with work internationally, but require me to pickup mostly from mainland China, or on occasions some other destination. Normally it goes like this:
1. I get an sms, specially when in the middle of the night for a request (Like: "Hi, you free for delivery 2mrw afternoon, PEK-AMS, open return. Call me asap please"
2. If I can and want to take it I call back immediately to work out the details (any choice of airline, how many items, delivered to agent or client, who does custom clearance)
3. I usually travel very light, so I have a bag packed for that purpose actually, so I just try and work out what I can do at the destination, who I could visit, and when I need to be back
4. The logistics company books the flights on my behalf, so I dont have to do that, except in a few exceptions, when it was easier to do for me (domestic China)
5. I head to the airport, meet the agent, take the items with the agent to customs (and I usually ask to see the contents) get the clearance papers and checkin
6. If its only documents, then i just take it as cabin luggage, if industrial parts/computer parts then its often checked-in.
7. I board and fly to destination, and either clear customs myself, or meet an agent at the baggage claim, who takes care of it.
8. If there is no agent, I once had to delivery it across town in Wuhan, nearly missed my outgoing flight, as I didnt want to stick around there on that occasion.
Thats a typical scenario. Now I hear some of you ask, why would a company spent all that money for a flight when there is in fact Fedex, DHL, SF Express, etc... The answer is both simple and suprising. None of these companies offer such a service, as its often highly customised. Imagine you run a fair sized production facility. While producing 20000 units of whatever a day, your controller system malfunctions, and you need a specialised chip or part that is (most often now) manufactured China, Japan, Taiwan (replace with any other country if you like). You now have the choice of waiting 3 days (more if its the start of a weekend/holiday) to have that replacement part. Thats a lot of days of lost production (and perhaps incurring penalties on top for missed deliveries). Hence a courier that costs them perhaps 2000-3000 USD in total is cheap.
Likewise, you may have a important person that needs a new passport, visa, permit, etc or a business deal which needs documents. Anything that is time sensitive, or of great value (or both). Lastly, the most important part if you looking to do this: be a reliable, fast communicator, have an unlimited passport, no travel restrictions, be reachable day and night, and perhaps most importantly, be based in a country that has little English speakers, and strict visa requirements. China is such a place, and I speak some mandarin, Russia would also be a good base, but China is unique in such, that it is also the origin of many products these days.
And, yes, I get paid, but its not much, and some pay by trip, some by days, or items. And before you ask, no - I will not give you the name of the companies, but if you looking for way to get to fly for free, its not much use, as I mentioned before, trips are irregular, and there is no choice where you go. You just choose, yes, or no!
Any more questions, I can try and answer, but I wont get into personal or company details.