Random projections recently became popular tools to process big data. When applied to Nonnegative Matrix Factorization (NMF), it was shown that, in practice, with the same compression level, structured random projections were more efficient than classical strategies based on, e.g., Gaussian compression. However, as they are data-dependent, they remain costly and might not fully benefit from recent very fast random projection techniques. In this paper, we thus investigate an alternative framework to structured random projections—named random projection streams (RPS)—which (i) are based on classical random compression strategies only—and are thus data-independent—and (ii) can benefit from the above fast techniques. We experimentally show that, under some mild conditions, RPS allow the same NMF performance as structured random projection along iterations. We also show that even a CPU implementation of Gaussian Compression Streams allows a faster convergence than structured random projections when applied to weighted NMF.