Try not to badger them about it and give them the time they need to talk about on their own.
Don't be too demanding. It's like catching a fly in water. You can't hit it with the spoon, you have to lay the spoon next to it gently and let it fall in.
Hmm...good question. I once had a co-worker who was very private about her personal life. Myself and another co-worker would ask a seemingly innocent question, and she would refuse to answer. My other co-worker would pester and badger her to death,asking her over and over, questioning her asbto what the big deal was. I, on the other hand, would not mention it, and would just go on as if it didn't matter. Eventually if it came up she would end up telling me. Though I do remember she would not tell me her birthday, telling me to pick a day, which I did. But she never did tell me. Maybe just not make a big deal about it, other than that I don't know.
Let them tell you in there own time unless you think it's an emergency.