Identifying characteristic vibrational modes and frequencies is of great importance for monitoring the health of structures such as buildings and bridges. In this work, we address the problem of estimating the modal parameters of a structure from small amounts of vibrational data collected from wireless sensors distributed on the structure. We consider a randomized spatial compression scheme for minimizing the amount of data that is collected and transmitted by the sensors. Using the recent technique of atomic norm minimization, we show that under certain conditions exact recovery of the mode shapes and frequencies is possible. In addition, in a simulation based on synthetic data, our method outperforms a singular value decomposition (SVD) based method for modal analysis that uses the uncompressed data set.