I am a psychiatry resident so right now I am not seeing many people who have tragic physical illnesses (although Psych people can go into Palliative Care fellowships - I'll get into that more below). However, general psych work is sad in a different way.
In psychiatry, you routinely have patients who are talking to you about things like terrible physical/emotional/sexual abuse, loneliness, homelessness, poverty, devastating personal losses (a few times on the inpatient unit we had parents who came in suicidal because their child had died and they just couldn't make peace with their overwhelming grief...those cases always were especially poignant to me). Then there are the psychotic patients who stay out in the cold weather and get frostbite or have suffered other maladies because they couldn't take care of themselves. It is heartbreaking if you really dwell on it.
There are a lot of times when at the end of the day, I do have a passing feeling of, "Whoa, what a day" because I had a couple of different patients with very intense, sad stories. Sometimes I do think about patients when I'm off from work because something will remind me of them. But when I'm off, it's normally just a passing thought and not something that haunts me. For my sanity, I need a break from it.
I do agree with the view that you deal with so many sad and tragic cases that you really can't take every one of them personally. Back during med school and internship, I would read the obituaries regularly. After every inpatient internal medicine rotation, it was inevitable that I'd see a LOT of familiar names in the obituaries for months after the rotation. A huge portion of IM inpatients are sick elderly people who are not long for this world.
Some patients do stick with me. I remember that it affected me more than I thought it would the first time that *I* was the one who told a patient that she had cancer (lung cancer that had already spread - and the patient had no suspicion of anything that serious).
Now, I wouldn't say I felt guilty or responsible in any way for the poor woman's situation. It's just a feeling of sadness and empathy, realizing that *you* are going to change someone's life with terrible news.
On the other hand, I actually did a palliative care rotation once, and contrary to what you might think it really wasn't a depressing experience. It was pretty amazing seeing how the palliative care doctor would go into these tragic situations trying to put the patient/family at ease and focusing on what we could do to make a terrible situation more peaceful, dignified, and humane. I absolutely think that palliative care is under-utilized in a lot of these sad scenarios.