Just curious because this question comes up often. I also got this question a lot when I was secretary for a Visiting Nurse Assoc. I live in South Jersey. When a person is hospitalized and needs rehab the Hospital handles it. They have a special employee that calls around to see where beds are available and then gives the patient or family members a choice. At no time have I had to call around asking if an opening is available. Homecare I picked which one I preferred, but the hospital called the service and sent them the doctor order. Then Homecare calls me to set up a day to admit my Mom. As a secretary I got lots of calls from family members freaking out because they thought the responsibility would be theirs. I explained that a Social Worker would contact them with how discharge will be handled. As a VNA facility we did nothing until we had a doctors order in hand from the hospital. Isn't this how all hospitals work?
From them on I merely asked for a list, then called, made arrangements for a tour and meeting, then told the discharge planner which ones interested me. I'm not sure she appreciated that, but I wasn't going to be limited to her choices, or whatever choices she was advised to use by someone above her in rank.
I have been told a few times that there might not be a bed available, so sometimes it was touch and go, but always a bed opened up so we got in where we wanted from on, except when I made a big mistake and chose a SNF which was "bottom of the barrel" in caliber.
I think who takes the lead on finding a place depends on the experience of the family.
I am in a large metro area so the Social Work had to jump at the opening as openings were far and few between. Couple times I was able to stall my parent's release and wait for another bed opening to a place closer to home. I couldn't stall too long as the health insurance would not pay for another day in the hospital.