Challenge: The Gospels Aren't Eyewitness Accounts

Here's the kind of question Jim often hears:

Christians like to call the Gospels "eyewitness accounts," but why should we take them seriously as "eyewitness" testimony when the authors don't say anything about being eyewitnesses and the accounts clearly contain descriptions of events (like the virgin conception, birth narratives, and private moments of Jesus) that the authors could not have personally seen?

So what do you think? Do you have an answer for this challenge? Let us know in the comments below, then Jim will post his answer on Thursday.