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I had a similiar problem when, many years ago, when I became a child abuse investigator. You think reporters ask impertinent questions--how about asking someone if he is having sex with his daughter? I was pretty shy and had grown up in a small town in Nova Scotia before moving as a teenager to a small town in Florida and asking questions like this were just not part of my personal culture.
However, it was my job and I had to do it. The first thing I did was to create the persona of "Ann the Investigator", a self assured, confident woman who had the legal and moral right to talk to people about the intimate details of their lives. "Ann the Investigator" was not the same quivering wreck as "Ann the Person".
Then I realized that it was not my job to be "Ann the Human Lie Detector Machine". My job was to accept what I was told as what the person was telling me, using that data to figure out what further questions to ask and where to get corroborating or refuting evidence.
Then I realized that it was all right to like my clients, even if they may have done terrible things. I could like them without liking what they did and without believing they were truthful.
I also discovered that there is absolutely nothing people like as much as talking about themselves and if you are interested and make encouraging noises,it is amazing what they will tell you. Now, with so much confessional television lowering the privacy bar, most people will start talking and will not stop until they have told you everything. It bogles the mind.
You can do it. You became a journalist because you were interested in people and in their stories. Given half a chance, they will tell you everything you want to know. You are not a prying jerk--you are a confessional facilitator.