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Nonnegative matrix factorization (NMF) has been successfully applied in several data mining tasks. Recently, there is an increasing interest in the acceleration of NMF, due to its high cost on large matrices. On the other hand, the privacy issue of NMF over federated data is worthy of attention, since NMF is prevalently applied in image and text analysis which may involve leveraging privacy data (e.g, medical image and record) across several parties (e.g., hospitals). In this paper, we study the acceleration and security problems of distributed NMF. Firstly, we propose a distributed sketched alternating nonnegative least squares (DSANLS) framework for NMF, which utilizes a matrix sketching technique to reduce the size of nonnegative least squares subproblems with a convergence guarantee. For the second problem, we show that DSANLS with modification can be adapted to the security setting, but only for one or limited iterations. Consequently, we propose four efficient distributed NMF methods in both synchronous and asynchronous settings with a security guarantee. We conduct extensive experiments on several real datasets to show the superiority of our proposed methods. The implementation of our methods is available at https://github.com/qianyuqiu79/DSANLS.