Passing something down orally makes it easy to change, yeah.
The gospels (Matthew, Mark, Luke, John) were not passed down orally. They were written shortly after the death of Christ, and copies were passed to the churches in different regions where Christianity was first starting out. One church might have a copy of John, and another might have a copy of Luke. They would also have letters from Paul, or copies of letters from Paul. (Rome, Corinth, Ephesus, etc.)
It's also important to remember that the apostles that actually knew Jesus went out and spread the word. I don't know if they had any idea how major Jesus would be two thousand years later. They seriously may have expected Christ to return during their lifetimes, or shortly thereafter. ( I wasn't them.
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Glyde Bane hit the nail on the head as far as what was written. They were eye witness accounts, and letters written from one apostle to a church or group. It wasn't until later that someone collected the letters and accounts to make an official book.
I'm not sure (knowledgeable) about corrupt Popes and censorship playing much of a role in anything, since modern translations of the Bible use the best manuscripts available from very long ago. I find it hard to believe that there could be a drastic difference between the Biblical account and what actually happened due to an intentional mistranslation by the church. The manuscripts are really consistent, so... I dunno about that.
The apostles went to different areas and preached what they believed to be true. If I'm not mistaken, tradition says that they all died for what they believed to be true. (12 disciples + Paul)
Being a martyr shows complete sincerity, and in the case of the apostles, it holds weight for my personnal opinion. I'm just talking about the resurrection, not necessarily the entire Bible.