Random projections belong to the major techniques to process big data and have been successfully applied to Nonnegative Matrix Factorization (NMF). However, they cannot be applied in the case of missing entries in the matrix to factorize, which occurs in many actual problems with large data matrices. In this paper, we thus aim to solve this issue and we propose a novel framework to apply random projections in weighted NMF, where the weight models the confidence in the data (or the absence of confidence in the case of missing data). We experimentally show the proposed framework to significantly speed-up state-of-the-art NMF methods under some mild conditions. In particular, the proposed strategy is particularly efficient when combined with Nesterov gradient or alternating least squares.