Although whatever Jesus said in his sermons was most likely in Aramaic it would have been written in Greek by people fluent in both.  Greek had been the "international" language for several centuries, since Alexander conquered the area and was used by anyone considered "educated" for business, legal or literary texts.  Even personal correspondence would have been written in Greek precisely to avoid imprecise translation.  The xenophobic Romans learned Greek as part of their education not just because of their cultural heritage but for the practical reason that it was the language of administration in "The East"  --  the most important part of their empire.

If things were altered in translation it's more likely to have happened centuries later when copyists who were not fluent in Greek tried to translate the Greek texts into Latin.

The Greek of the time was perfectly capable of transmitting the subtleties of theology.  The original schism in the early church concerning the exact natures of God the father, God the son and the relationship between them were debated at the conclaves of Nicea and Chalcedon in Greek, presided over by Roman Emperors who took part in the debates.

What went into the New Testament, or was left out, was quite possibly political or just what suited whoever was compiling the book at the time but the original Greek texts were probably a reasonably accurate record of what was actually said.