I've hardly had a problem, nor seen many providers have a problem with the patient's depressing stories. IMHO because we're working to get these people better, and that emotion overrides the feeling of sorrow for them.
I got home an hour late today because a patient of mine got tremendously better, to the point where I was able to cut down his tx costs to the point where he has an extra $500 a month, and he's in a position where he can now find work (before he was pretty much unable to work). That $500 for this guy goes a long way. He can now actually buy decent food for his family, and he's got two beautiful kids. The guy called me up, telling me I really helped him out, he's doing to drop his quest to get SSDI and it was a good feeling for me. The entire time I had this guy up until this point, he did have a pretty sad story, but never did it make me feel terrible to the point where it took me over because I knew I was helping him and that he realistically could get out of his bad situation.
(>3.8 GPA, developed severe back problems, then later opioid dependence because of a quack doctor that kept giving him ever higher amounts of opioids, he later developed panic disorder and atrial fibrillation, had to drop out of school, and couldn't work due to the back pain. Got him on Suboxone, he's been weaned off of it, back pain is for the most part under control due to a good pain management doctor, a real one, not a pill-pusher, panic attacks are now down to about once a month).
The only times I see providers consistently really feel bad because of the woes of their patients is when a suicide happens because then the provider cannot get the patient better, and they go into a state of pseudo-PTSD, becoming too safe and over-questioning everything they do for a few weeks.
A colleague of mine that has several more years of experience and IMHO is superior to myself as a psychiatrist had a suicide a few months ago, and I saw him too go through the same thing mentioned above. Felt odd to see someone better than me and with more experience default to feeling insecure for a few weeks.
What I see much more often that's the problem with counter-transference are when we get mad at patients, and these usually happened with malingering, cluster B or C personality disorders, or factitious disorder.
When I did private practice, I had some counter-transference issues I noticed didn't happen until then. It was a time where the majority of my patients were people I identified with because they were almost all middle class or higher. When you work in the inner-city, you often get homeless patient with a wrap sheet of several misdemeanors and/or felonies. In private practice, you get people that could've been your neighbor.
So on occasion, I had very attractive women come to me for help, several of them for sexual issues and immediately I noticed myself thinking "don't even go there," and I kept that thought up the entire time like a force-field. I never had that problem ever with any patients I previously had.