I was on a list of people who did not have family doctors with a local branch of a government agency. When my doctor became available, I received a letter with an appointment time. I automatically chose to go with him - I could have chosen not to and gone back on the list. The only reason I got the initial appointment time was because my status was upgraded based on a change in my health. Within a couple days, I got the letter. It was amazingly quick.
My family members, on the other hand, did some shopping around. When our previous family physician retired, they spent the time to find physicians that were taking new patients and went to one doctor. They committed. They all hated her after a few months, and eventually they all moved to another doctor they found that they really like.
My mom got to choose her new psychiatrist. I kept dismissing psychiatrists as a teenager until I found one that I liked, but it was about 9 months between each until there would be an opening in their schedules. But, there are some differnences in adult and adolescent psychiatry in the region I am from.
I did not get to choose my neuro-ophthalmologist because there is only a handful in North America and I had to travel an hour by car to get to him. I did not get to choose my neurosurgeon. I did choose my ophthalmologist and optometrist. I did get some choice with my general surgeon. I more or less picked my cardiologist and endocrinologist, who both had clinics and were completely independent of the hospital system.
With most specialists, my family doctor would make the referral to the individual he thought was most helpful and I think he also considered who I wouldn't blow off (meaning, who I would like or at least not dislike).