A better statement would be: if two rational agents share the same prior beliefs and their posterior beliefs are common knowledge, irrespective of what private information they have been exposed to, they cannot rationally agree to disagree.

When are beliefs common knowledge? when both agents can directly observe one another's beliefs. I.e. Bob must know Alice knows that Bob knows that .... ad inifinitum that XYZ is true. Mutually witnessing an event is sufficient for common knowledge.

I feel this is not a useful day-to-day heuristic since the theorem was intended to highlight deficiencies within the Bayesian rational paradigm (specifically the common prior assumption since game theorists weren't ready to abandon rationality in the 70's).