There isn't one model for this. Generally, if the hospital is your employer, they cover you. If a group is your employer, the group pays. If you are moonlighting in a hospital, but your job or residency is elsewhere, sometimes you have to self cover.
Medical malpractice affects your bottom line income, no question. And every physician can expect to be sued at least once in their career. In some fields (OB, surgery) you can expect to be sued more often than that. And in most of these cases, some mistake will have been made. You absolutely will make mistakes. Some of them more costly than others. Nobody is perfect and when you take care of hundreds to thousands of people a year, you will miss something in a diagnosis, you will cut something important during a surgery, you will administer a treatment that makes somebody worse off, and in retrospect you should have known it. It happens. Which is why doctors have so much coverage. The bigger the damage you can do, and the more likely, the more you want to be insured. Most look at it as a cost of doing business. In fact many other fields carry malpractice insurance -- this isn't just a doctor issue. eg lawyers do, sometimes at much higher levels of coverage than doctors, depending on the deal sizes they deal with. I've worked on deals where the firm upped their per lawyer insurance to far higher amounts than any doctor would ever carry, because if a million or billion dollar deal gets screwed up somebody has to pay. That too was just a cost of being in this lucrative business.
A relatively recent study from Harvard med suggested that the percentage of lawsuits that are frivolous is really not that great. Thus medmal insurance is serving a valid purpose of covering folks when errors actually are made. It is an unfortunate consequence of a society which doesn't really understand medicine or science and thinks doctors are infallable and as brilliant as the characters on their favorite doctor TV show. The expectations are high. But the opportunities to screw up somewhere are higher.
As for OB being "wiped out", this was a concern in a couple of regions, but hasn't really been shown to be the case. To some extent it was overblown by the media. Lots of people go to where the money is, and so when OB gets less lucrative because of medmal costs, logically folks with advanced degrees who can earn more elsewhere tend to do so, so you were seeing some folks in some regions trying to focus on the GYN aspects or switching to other areas of medicine. And a lot of people who had no intention of switching, but did a good job of whining to the press about how they were being sued into oblivion. In fact OB is not close to going out of business, is a very strong program at a lot of med schools, and you are seeing more and more encroachment by family medicine which is also a field that sometimes delivers babies (and without the specific OB expertise probably requires even higher medmal coverage to do so)..
