I am certainly not an expert, but there are a few things you should follow to find your job security.
Most private hospitals contract for anesthesia service to a company (the company is usually owned by MD's, CRNA's, or perhaps a large organization that is owned by neither). It is then the company's job to provide qualified people, be they MD's, CRNA's, AA's, etc. Usually, they provide coverage for the hospital in order to gain the privledge of billing all of the patients. If the hospital has a lot of uninsured patients or for some other reason the company can't make money, the hospital may allow the anesthesia company to bill on its own AND provide a small supplement out of the hospital profit, but that is not normal.
You can get a job with the company that is providing the services, you can sign a "contract" with them for as long or as little as you want (here in Texas, we have what is called "at will" employment meaning the employee or employer can terminate the employment relationship at any time). If the company loses the contract or goes belly up, you will be laid off or asked to relocate, even if you have time left on your contract, so if you are looking for security, see how well run the company is run.
Locum contracts are a little more stable, but are usually for a shorter period of time. Hospitals very rarely change providers on something as big as anesthesia, though it does happen.
If you work for a university or a large learning hospital, you may be able to get a very long term job. If you are teaching, you may even be able to get tenure, but you will most likely have lower pay (due to the teaching status).
I hope this helps to answer your question.