We present Magellan/IMACS, Anglo-Australian Telescope/AAOmega+2dF, and Very Large Telescope/GIRAFFE+FLAMES spectroscopy of the Carina. II ( Car II) and Carina. III ( Car III) dwarf galaxy candidates, recently discovered in the Magellanic Satellites Survey ( MagLiteS). We identify 18 member stars in Car. II, including two binaries with variable radial velocities and two RR Lyrae stars. The other 14 members have a mean heliocentric velocity v(hel) = 477.2 +/- 1.2 km s(-1) and a velocity dispersion of sigma(v) 3.4(-0.8)(vertical bar 1.2) km s(-1). Assuming Car II is in dynamical equilibrium, we derive a total mass within the half-light radius of 1.0(-0.4)(+0.8) x 10(6) M circle dot, indicating a mass-to-light ratio of 369(-161)(+309) M circle dot/L circle dot. From equivalent width measurements of the calcium triplet lines of nine red giant branch (RGB) stars, we derive a mean metallicity of [Fe/H] = -2.44 +/- 0.09 with dispersion sigma([Fe/H]) = 0.22(-0.07)(+0.10). Considering both the kinematic and chemical properties, we conclude that Car II is a dark-matter-dominated dwarf galaxy. For Car III, we identify four member stars, from which we calculate a systemic velocity of vhel = 284.6(-3.1)(+3.4) km S-1. The brightest RGB member of Car. III has a metallicity of [Fe/H] = -1.97 +/- 0.12. Due to the small size of the Car III spectroscopic sample, we cannot conclusively determine its nature. Although these two systems have the smallest known physical separation (Delta d similar to 10 kpc) among Local Group satellites, the large difference in their systemic velocities, similar to 200 km s(-1), indicates that they are unlikely to be a bound pair. One or both systems are likely associated with the Large Magellanic Cloud (LMC), and may remain LMC satellites today. No statistically significant excess of gamma-ray emission is found at the locations of Car II and Car III in eight years of Fermi-LAT data.