The spectroscopy of background QSO sightlines passing close to foreground galaxies is a potent technique for studying the circumgalactic medium (CGM). However, QSOs are effectively point sources, limiting their potential to constrain the size of circumgalactic gaseous structures. Here we present the first large Keck/Low-resolution Imaging Spectrometer (LRIS) and Very Large Telescope (VLT)/Focal Reducer/Low-dispersion Spectrograph 2 (FORS2) spectroscopic survey of bright (B_AB_<22.3) background galaxies whose lines of sight probe MgII{lambda}{lambda}2796,2803 absorption from the CGM around close projected foreground galaxies at transverse distances 10kpc<R_{perp}_<150kpc. Our sample of 72 projected pairs, drawn from the PRIsm MUlti-object Survey, includes 48 background galaxies that do not host bright active galactic nuclei, and both star-forming and quiescent foreground galaxies with stellar masses of 9.0<logM_*_/M_{sun}_<11.2 at redshifts of 0.35<z_f/g_<0.8. We detect MgII absorption associated with these foreground galaxies with equivalent widths of 0.25{AA}<W_2796_<2.6{AA} at >2{sigma} significance in 20 individual background sightlines passing within R_{perp}_<50kpc and place 2{sigma} upper limits on W_2796_ of <~0.5{AA} in an additional 11 close sightlines. Within R_{perp}_<50kpc, W_2796_ is anticorrelated with R_{perp}_, consistent with analyses of Mg ii absorption detected along background QSO sightlines. Subsamples of these foreground hosts divided at logM_*_/M_{sun}_=9.9 exhibit statistically inconsistent W_2796_ distributions at 30kpc<R_{perp}_<50kpc, with the higher-M* galaxies yielding a larger median W_2796_ by 0.9{AA}. Finally, we demonstrate that foreground galaxies with similar stellar masses exhibit the same median W_2796_ at a given R_{perp}_ to within <0.2{AA} toward both background galaxies and toward QSO sightlines drawn from the literature. Analysis of these data sets constraining the spatial coherence scale of circumgalactic MgII absorption is presented in a companion paper.