Uncertainties are distributions, not errorbar values

This clicked on me while preparing for a talk, though it's an obvious thing: too often we are content with just plotting errorbars and not specifying what do they mean (for instance: 1 sigma-uncertainties, Gaussian distribution). This makes propagating uncertainties (and using the data for model selection) actually impossible.
To put it another way: we don't have datapoints, we operate with slippery, diffuse clouds of probabilities of data. My current task is to take full 2D uncertainties distribution for my datapoints and re-use them in the following steps of hierarchical analysis.