Diagnosability is the property of a given partially observable system model to always exhibit unambiguously a failure behavior from its only available observations in finite time after the fault occurrence, which is the basic question that underlies diagnosis taking into account its requirements at design stage. However, for the sake of simplicity, the previous works on diagnosability analysis of discrete event systems (DESs) have the same assumption that any observable event can be globally observed, which is at the price of privacy. In this paper, we first briefly describe cooperative diagnosis architecture for DESs with autonomous components, where any component can only observe its own observable events and thus keeps its internal structure private. And then a new definition of cooperative diagnosability is consequently proposed. At the same time, we present a formal framework for cooperative diagnosability checking, where global consistency of local diagnosability analysis can be achieved by analyzing communication compatibility between local twin plants without any synchronization. The formal algorithm with its discussion is provided as well.