This goes through all definitions of entropy, except the very first one, which is also the one that is in fact measurable and objective: the variation in entropy is the amount of heat energy that the system exchanges with the environment at a given temperature during a reversible process. While tedious, this can be measured, and it doesn't depend on any subjective knowledge about the system. Any two observers will agree on this value, even if one knows all of the details of every single microstate.