One of the common mistakes that evangelists make is in using the bible as proof of its own authority. While this is acceptable when the common ground of faith has been established, it is not acceptable in outreach.
When you are debating a person, the first thing you should do is ascertain which sources are considered credible by the person you are debating. Much of the Old Testament would be acceptable in discussion with a Jew or Muslim, but the New Testament is not common ground (it would help in both of these instances, however, if you knew their OTHER scriptures well as well). Just as you would not accept Darwin's "Origin of Species" as a proof text, neither should your opponent be asked to accept the bible as a proof text without a great deal of support that most Christians are unable to provide.
As an example, I will use this. As some of you well know I was raised Mormon. I do not agree with the doctrine of the LDS church, and, as a young Christian attempted to witness to Mormons at every opportunity. When I began doing my research to witness, I found that much of the information gleaned was from ex Mormons who tended to exaggerate and lie about the faith because they had an agenda to push (the film "The Godmakers" is an appalling example of this kind of hyperbole). While I found some of the information true and useful, most was rubbish, and so I found myself referencing the scriptures of the LDS church (my bet is that MOST evangelists can't name the key scriptures, listed here for your convenience and my own "street cred": The Book of Mormon, the Doctrine & Covenants, the Word of Wisdom, the Pearl of Great Price, and the Holy Bible "insofar as it is translated correctly"), as well as LDS authored history books and writings so that my discussion could center on a KNOWLEDGE of their faith rather than parrotting the words of other unbelievers in the LDS faith.
An embarrassing incident arose about ten years ago from evangelists accepting someone's word about another faith without examining the evidence. Mike Warnke was a prominent evangelist who claimed to have once been a satanist high priest. Much of the "information" evangelists gathered on Satanism came from Mike's "testimony", which was later proven to be composed of sensationalist lies.
The nonChristian realizes this, and is reluctant to accept the Bible on its own as proof of its own veracity. And so, the Christian who wishes to witness to the unbeliever must rely on works acceptable to the person to whom they are witnessing rather than a book that not all accept to be divinely inspired.