Of course, they can say "no" - you can't force anyone.
The way it works is that say, you have a client for an appointment tomorrow at 5, which you can't make, so you call your self-employed therapist Jane and ask: "can you do a facial tomorrow at 5 at x area?" Jane says: "no, sorry, I am fully booked, but I could do the next day." then you call your 2nd Therapist, Stacy and repeat the question. Stacy says: "Yes, I can." and you book her in. If the 2nd Therapist can't make it either, try to negotiate the appoitment time with the client - maybe you still book her in for the next day with Jane. Or have more Therapists in your books.
All the rates are must be agrred on in adavance. Then you get the client to prepay the appointment and explain your cancellation policy, as you don't want to pay your Therapist for the time wasted from your pocket in case the client is not in or changes their mind last minute.
After the appointment is done, you transfer the Therapist's share to their account.
This is casual job - they might have 1-2 appointment from you a day or maybe even a week. And of course they will juggle it with their own clients, jobs and committments.
Trade-testing and interviewing is crucial here. I had so many desperate people on the phone when advertising for extra specialists, and guess what? The most desperate never turned up for an interview. Surely they wouldn't make it to the client either.
In my opinion it is all about building the network and communication and I would say, it is well worth it.
This is also the way leading mobile companies (or agencies if you prefer) work.