Photometry alone is not sufficient to unambiguously distinguish between ultra-faint star clusters and dwarf galaxies because of their overlap in morphological properties. Here we report on VLT/GIRAFFE spectra of candidate member stars in two recently discovered ultra-faint satellites Reticulum 2 and Horologium 1, obtained as part of the ongoing Gaia-ESO Survey. We identify 18 members in Reticulum 2 and 5 in Horologium 1. We find Reticulum 2 to have a velocity dispersion of ~3.22 km/s, implying a M/L ratio of ~ 500. We have inferred stellar parameters for all candidates and we find Reticulum 2 to have a mean metallicity of [Fe/H] = -2.46+/-0.1, with an intrinsic dispersion of ~ 0.29, and is alpha-enhanced to the level of [alpha/Fe]~0.4. We conclude that Reticulum 2 is a dwarf galaxy. We also report on the serendipitous discovery of four stars in a previously unknown stellar substructure near Reticulum 2 with [Fe/H] ~ -2 and V_hel ~ 220 km/s, far from the systemic velocity of Reticulum 2. For Horologium 1 we infer a velocity dispersion of 4.9^{+2.8}_{-0.9} km/s and a consequent M/L ratio of ~ 600, leading us to conclude that Horologium 1 is also a dwarf galaxy. Horologium 1 is slightly more metal-poor than Reticulum 2 [Fe/H] = -2.76 +/- 0.1 and is similarly alpha-enhanced: [alpha/Fe] ~ 0.3. Despite a large error-bar, we also measure a significant spread of metallicities of 0.17 dex which strengthen the evidence that Horologium 1 is indeed a dwarf galaxy. The line-of-sight velocity of Reticulum 2 is offset by some 100 km/s from the prediction of the orbital velocity of the LMC, thus making its association with the Cloud uncertain. However, at the location of Horologium 1, both the backward integrated LMC's orbit and the LMC's halo are predicted to have radial velocities similar to that of the dwarf. Therefore, it is likely that Horologium 1 is or once was a member of the Magellanic Family. ; Comment: submitted to ApJ, abstract abridged