We present Spitzer 7-38um spectra for a 24um flux limited sample of galaxies at z~0.7 in the COSMOS field. The detailed high-quality spectra allow us to cleanly separate star formation (SF) and active galactic nucleus (AGN) in individual galaxies. We first decompose mid-infrared Luminosity Functions (LFs). We find that the SF 8um and 15um LFs are well described by Schechter functions. AGNs dominate the space density at high luminosities, which leads to the shallow bright-end slope of the

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... mid-infrared LFs. The total infrared (8-1000um) LF from 70um selected galaxies shows a shallower bright-end slope than the bolometrically corrected SF 15um LF, owing to the intrinsic dispersion in the mid-to-far-infrared spectral energy distributions. We then study the contemporary growth of galaxies and their supermassive black holes (BHs). Seven of the 31 Luminous Infrared Galaxies with Spitzer spectra host luminous AGNs, implying an AGN duty cycle of 23+/-9%. The time-averaged ratio of BH accretion rate and SF rate matches the local M_BH-M_bulge relation and the M_BH-M_host relation at z ~ 1. These results favor co-evolution scenarios in which BH growth and intense SF happen in the same event but the former spans a shorter lifetime than the latter. Finally, we compare our mid-infrared spectroscopic selection with other AGN identification methods and discuss candidate Compton-thick AGNs in the sample. While only half of the mid-infrared spectroscopically selected AGNs are detected in X-ray, ~90% of them can be identified with their near-infrared spectral indices.