Do the speeches in the Bible record the exact words of Peter, Paul, or Jesus? And if not, why should we trust them?
In ancient times, transcription from dictation was not the method used to convey speech. It is pretty clear that the speeches, like Peter's Pentecost message in Acts 2, or Paul's Areopagus address in Acts 17, are greatly condensed. If you are unsure about this, try reading them aloud, and then ask yourself if the apostles--or Jesus, for that matter--made it a habit of speaking no longer than 2-3 minutes at a time.
The speeches are condensed and paraphrased. Biblical writers did not feel a burden to report the exact words when they recorded a conversation. In fact, everything Jesus said that is recorded in the Bible can be read aloud in just a couple of hours. Surely Jesus spoke for more than two hours during his three-year ministry, but how fortunate it is that the Bible didn’t record everything he said—the gospels alone would be tens of thousands of pages long.
An ancient historian illuminates the ancient method. Thucydides of Athens (460-395 BC), in his Inquiries (usually rendered Histories), 1.22.1, explains: As to speeches that were made… it was hard for me, and for others who reported them to me, to recollect the exact words. I have therefore put into the mouth of each speaker the sentiments proper to the occasion, expressed as I thought he would be likely to express them, while at the same time I endeavored, as nearly as I could, to give the general sense of what was actually said.
Several centuries later, a Christian leader, Papias of Hierapolis (writing around 125 AD), relates: Mark, having become the interpreter of Peter, wrote down accurately whatever he remembered. It was not, however, in exact order that he related the sayings or deeds of Christ. For he neither heard the Lord nor accompanied Him. But afterwards, as I said, he accompanied Peter, who accommodated his instructions to the necessities [of his hearers], but with no intention of giving a regular narrative of the Lord’s sayings… He took special care not to omit anything he had heard, and not to put anything fictitious into the statements.
Papias notes that the gospel of Mark is a truthful account, even if not strictly chronological. This may offend modern journalistic or evangelical sensibilities, but it was standard among ancient writers of biography.
For more on this matter, please see A Quick Overview of the Bible, from which I have made use of a few excerpts.