We address issues related to privacy protection in distributed information sharing systems where multiple autonomous entities share data across their private databases. Most existing solutions place restrictions on adversarial behavior in order to enable communication-efficient privacy-preserving information sharing. These restrictions substantially underestimate the capabilities of adversaries in reality, and do not always suffice for real systems. We consider a threat space containing more powerful adversaries, including not only semihonest but also malicious ones, and analyze the tradeoff between privacy protection and communication complexity in information sharing. In particular, we use Kolmogorov complexity to derive lower bounds on the communication complexity required to defend against various kinds of adversaries.