If you look back to see how people have previously described this encounter, they have usually seen it in the light of a philosophical or theological argument. Yet in the ancient world, these matters were never just theological concepts. If it were some kind of an intellectual discussion, why should Sherem end up dying over it?
There are clues that we are in the sphere of high-level legal material from the very outset. That word "contend" shows up in the Old Testament, so we have a good idea what it would have been in Hebrew; it is rib, the word for a "lawsuit." So, when it says that Sherem contends, it means that he was bringing a legal action. He was raising an accusation against Jacob. As the high priest, Jacob would typically have been the chief judge in such matters. However, since Jacob was the accused, all that either of them could do was submit the matter to divine judgment. Likewise, in this context, if Jacob bore to Sherem his "testimony," the word would mean here "witness," as in a witness in court under oath.
The accuser typically had to have personal knowledge of the alleged violation before he could commence his accusation. This is somewhat like our rule that you have to have standing in order to bring a lawsuit. The accuser also bore the burden of proof.
John W. Welch, "The Case of Sherem," The Legal Cases in the Book of Mormon, 122. "The crucial test in this regard is found in Deuteronomy: ‘If a false witness rise up against any man to testify against him that which is wrong; Then both the men, between whom the controversy is, shall stand before the LORD’ (Deuteronomy 19:16–17). Thus, Sherem’s conduct requesting Jacob to produce divine evidence was not a casual case of idle sign seeking, but rather followed a significant rule of ancient Israelite jurisprudence.’"