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This paper addresses the consistency issue of multi-robot distributed cooperative localization. We introduce a consistent distributed cooperative localization algorithm conducting state estimation in a transformed coordinate. The core idea involves a linear time-varying coordinated transformation to render the propagation Jacobian independent of the state and make it suitable for a distributed manner. This transformation is seamlessly integrated into a server-based distributed cooperative localization framework, in which each robot estimates its own state while the server maintains the cross-correlations. The transformation ensures the correct observability property of the entire framework. Moreover, the algorithm accommodates various types of robot-to-robot relative measurements, broadening its applicability. Through simulations and real-world dataset experiments, the proposed algorithm has demonstrated better performance in terms of both consistency and accuracy compared to existing algorithms.