Various assumptions on a right-censoring mechanism to ensure consistency of the Kaplan--Meier and Aalen--Johansen estimators in a competing risks setting are studied. Specifically, eight different assumptions are seen to fall in two categories: a weaker identifiability assumption, which is the weakest possible assumption in a precise sense, and a stronger representativity assumption which ensures the existence of an independent censoring time. When a given censoring time is considered, similar assumptions can be made on the censoring time. This allows for a characterization of so-called pointwise independence as well as full independence of censoring time and event time and type. Examples illustrate how the various assumptions differ.