Data anonymization is an approach to privacy-preserving data release aimed at preventing participants reidentification, and it is an important alternative to differential privacy in applications that cannot tolerate noisy data. Existing algorithms for enforcing $k$-anonymity in the released data assume that the curator performing the anonymization has complete access to the original data. Reasons for limiting this access range from undesirability to complete infeasibility. This paper explores ideas -- objectives, metrics, protocols, and extensions -- for reducing the trust that must be placed in the curator, while still maintaining a statistical notion of $k$-anonymity. We suggest trust (amount of information provided to the curator) and privacy (anonymity of the participants) as the primary objectives of such a framework. We describe a class of protocols aimed at achieving these goals, proposing new metrics of privacy in the process, and proving related bounds. We conclude by discussing a natural extension of this work that completely removes the need for a central curator.