Without having any idea of the nature of the "sensitive information" it difficult to answer, especially since what bothers one person might not annoy another.
To give you some an example- a great aunt (very old-fashioned elderly spinster) was the 1st person to mention we are descended from the 1st prostitute in New Amsterdam (and I was only 12 when she died). However, she didn't like to talk about a 'shameful' episode in the family history (a Revolutionary soldier in a family of Loyalists appointed Governor of New Jersey by George Washington).
Over the years I have asked relatives various questions about illegitimate children, bigamous marriages, suicides and other tragic deaths, prison terms, etc. but it depends on the person. Equally I haven't discussed the same issues with other relatives as I was fairly certain they would be upset by the subject.