I'm not sure what a doctors prescription does for this. I imagine the doctor is just a middle man in most CPAP journeys. As-in, the doctor is not the one who provides or trains you to use the CPAP. A third-party medical device provider does (Also, the training is a two minute, "put-the-mask-on and use distilled water, now sign here" kind-of-thing)
Edit: BTW I think the deadly thing is how long it takes to get a CPAP, from scheduling with a doctor to finally seeing them and getting a device it takes like 6mo to 1+ year. EVEN IF THEY THINK YOUR CASE IS VERY BAD. (I went to a sleep doctor in a hospital and they said I had one of the worst apnea cases he had seen, it still took another 3+mo from that point to get a device...)
You have 1. a device manufacturer, 2. a third party device seller, and 3. a sleep doctor. It takes time to get seen by a sleep doctor who then orders a sleep study device that needs to be provided by party 2 who gets it from party 1. Party 2 then provides results to party 3 who then prescribes a device, party 2 then needs to give you one of there devices if they have them in stock, if not they need to order them from party 1.
IMO there are two middle men here, the doctor and the device shop.
If you get a prescription you can get devices directly online (from sites who are just a more market-competitive version of party 2 above), but sometimes ResMed and Philips only sell there newer devices to people in the middleman scenario described above. E.g. for the first year or two of the ResMed AirSense 11, online shops (still requiring prescriptions) were not able to sell them.