The large volume of hyperspectral images (HSI) generated creates huge challenges for transmission and storage, making data compression more and more important. Compressive Sensing (CS) is an effective data compression technology that shows that when a signal is sparse in some basis, only a small number of measurements are needed for exact signal recovery. Distributed CS (DCS) takes advantage of both intra- and inter- signal correlations to reduce the number of measurements needed for multichannel-signal recovery. HSI can be observed by the DCS framework to reduce the volume of data significantly. The traditional method for estimating endmembers (spectral information) first recovers the images from the compressive HSI and then estimates endmembers via the recovered images. The recovery step takes considerable time and introduces errors into the estimation step. In this paper, we propose a novel method, by designing a type of coherent measurement matrix, to estimate endmembers directly from the compressively observed HSI data via convex geometry (CG) approaches without recovering the images. Numerical simulations show that the proposed method outperforms the traditional method with better estimation speed and better (or comparable) accuracy in both noisy and noiseless cases.