Well, I wrote a long response to this yesterday evening, but as it's not here it must have got lost in posting. 
The job I do is that of a Death Doula, but my actual title is Chaplain. I am very experienced in end of life care and offer support not only to the dying person, but their family, friends and any staff who care for them; after death has occurred I'm sometimes asked to take the funeral service too.
I'm a trained and qualified counsellor, which helps me to remain both compassionate and objective when I'm with the dying person and their families etc. I'm comfortable with talking about death and grief, I'm also comfortable with physical touch (many people like to have a hand held or hair stroked towards the end) and I understand the physical changes that often accompany the end of life experience. I am at peace with the process of dying.
I do occasionally go home and bawl my eyes out, but that's because I'm not actually an automaton - and because I can't do that in front of the family as I need to be present to help them through their grief, not process my own.
Sometimes (often in fact) there is unexpected humour. Yesterday I had a conversation, in a cafe, with a woman who wanted to discuss the arrangements for her own and her brother's deaths. We were deep in conversation about coffins when the waitress came over, and overhearing, she dropped our coffees on the table and skittered off like a scalded cat. 