I'm a Navy anesthesiologist. The answer is that it varies, but the risk is significant.
At the smaller hospitals - particularly overseas - the case load is dismal. The up side, at least for those stationed in CONUS, is that anesthesiology is very well suited to off-duty employment (moonlighting). We don't have ongoing responsibility for patients before or after they're in the OR, it's shift work, and there's a need for coverage at night and on weekends which can sometimes minimize the amount of leave one needs to take to do it. Most commands permit it with minimal hassle.
Anesthesiologists stationed at one of the large MTFs can generally practice something approaching the full spectrum of cases, albeit skewed heavily toward routine general surgery and orthopedics, with ENT urology OB/GYN etc thrown in. Case load is low. Not a lot of intracranial neurosurgery (but a good number of necks and backs), major vascular. No cardiac.
The lack of cardiac surgery is relevant to my interests, since less than a year after I finished an adult cardiac fellowship, my MTF closed its cardiac surgery program. I've been able to continue doing cases at a VA (on Navy time for my usual Navy pay) and at a local civilian hospital (on my time for locums pay). I was deployed just a few weeks ago, but until that point for 2019 I was on a pace to do somewhere around 75 pump cases, which isn't great but it certainly isn't cripplingly bad either. I'm aggressive with CME on my own time. I think I'm solid if not excellent
and don't think I'll have significant catchup or remediation to do when I leave the Navy and join a busy high acuity practice. But I'd be lying if I said that risk doesn't weigh on me a little.
I would say a Navy practice for an anesthesiologist is fine for a generalist. Case volume isn't great, but most of us moonlight and I know many many active duty anesthesiologists who've left the Navy after their time was up, or after retirement, and they all do Just Fine as civilians.
Our pediatric fellowship trained anesthesiologists are pretty busy. They have a good and varied case load. Whether that will be the case in 5 or 10 years, after we've shifted priorities to operationally relevant specialties and perhaps lost the peds and OB/NICU pipeline to the pediatric ORs ... is anybody's guess. I just don't know.
CCM fellowship trained anesthesiologists are pretty welcome in military hospital ICUs. A hybrid OR/ICU practice is something that can be tough to make happen in private practice, so this is perhaps OK in the military. The problem, of course, is that nearly all of our ICUs are very low acuity, low volume places. But CCM-trained anesthesiologists can probably at least keep their fingers in the ICU while they're on active duty. We have some anesthesiologists who do 100% CCM.
Pain fellowship trained anesthesiologists have plenty of work to do. I didn't like pain at all, and I don't really keep tabs on what they're doing. But I get the impression they have a pretty good practice. Good volume of procedures, good patient population, relatively low number of opioid-related headaches compared to many civilian practices.
Bottom line, it's worse than it was 10 years ago, but I still think it's OK, though my opinion is that moonlighting to supplement case volume ranges from merely important ... to absolutely critical (as in my own case).
As a final point, I will add that I burn the majority of my Navy vacation time moonlighting in order to maximize my case load. This is a recipe for burnout. I admit I was starting to feel it a little this summer, after a block of ~2 years in which I took almost no leave for personal time off. (I did have a ~3 month block of overseas TAD that was a welcome change of pace despite being somewhat traumatic for other reasons.) I left for deployment last month, and honestly had been looking forward to a little bit of time doing nothing.