In this paper, we propose a decentralized algorithm to solve the low-rank matrix completion problem and analyze its privacy-preserving property. Suppose that we want to recover a low-rank matrix \({ {D}} = [{ {D}}_{1}, { {D}}_{2}, \cdots , { {D}}_{L}]\) from a subset of its entries. In a network composed of \(L\) agents, each agent \(i\) observes some entries of \({ {D}}_{i}\). We factorize the unknown matrix \({ {D}}\) as the product of a public matrix \({ {X}}\) which is common to all agents and a private matrix \({ {Y}} = [{ {Y}}_{1}, { {Y}}_{2}, \cdots , { {Y}}_{L}]\) of which \({ {Y}}_{i}\) is held by agent \(i\) only. Each agent \(i\) updates \({ {Y}}_{i}\) and its local estimate of \({ {X}}\), denoted by \({ {X}}_{(i)}\), in an alternating manner. Through exchanging information with neighbors, all the agents move toward a consensus on the estimates \({ {X}}_{(i)}\). Once the consensus is (nearly) reached throughout the network, each agent \(i\) recovers \({ {D}}_{i} = { {X}}_{(i)}{ {Y}}_{i}\), thus \({ {D}}\) is recovered. In this progress, communication through the network may disclose sensitive information about the data matrices \({ {D}}_{i}\) to a malicious agent. We prove that in the proposed algorithm, D-LMaFit, if the network topology is well designed, the malicious agent is unable to reconstruct the sensitive information from others.