In this paper, we propose and study a distributed and secure algorithm for computing dominant (or truncated) singular value decompositions (SVD) of large and distributed data matrices. We consider the scenario where each node privately holds a subset of columns and only exchanges ``safe" information with other nodes in a collaborative effort to calculate a dominant SVD for the whole matrix. In the framework of alternating direction methods of multipliers (ADMM), we propose a novel formulation for building consensus by equalizing subspaces spanned by splitting variables instead of equalizing the variables themselves. This technique greatly relaxes feasibility restrictions and accelerates convergence significantly, while at the same time yielding simple subproblems. We design several algorithmic features, including a low-rank multiplier formula and mechanisms for controlling subproblem solution accuracies, to increase the algorithm's computational efficiency and reduce its communication overhead. More importantly, unlike most existing distributed or parallelized algorithms, our algorithm preserves the privacy of locally-held data; that is, none of the nodes can recover the data stored in another node through information exchanged during communications. We present convergence analysis results, including a worst-case complexity estimate, and extensive experimental results indicating that the proposed algorithm, while safely guarding data privacy, has a strong potential to deliver a cutting-edge performance, especially when communication costs are relatively high.