In my view, it's often even simpler than difference in assumptions in the more general sense: typically it simply comes down to people using the same words but having different definitions in mind for those words.
The assumptions can almost always be traced to some "fact" they believe to be true. And facts are easy to fake.
A good example is this recent Reddit thread based entirely on false "facts" promoted by the media, but believed to be true by significant chunk of the US population https://www.reddit.com/r/PoliticalHumor/comments/px9vr2/bles...
So the debate can often be reduced to something as basic as difference in sources you trust, rather than semantics.