We present deep Magellan/Megacam stellar photometry of four recently discovered faint Milky Way satellites: Sagittarius II (Sgr II), Reticulum II (Ret II), Phoenix II (Phe II), and Tucana III (Tuc III). Our photometry reaches ~2-3 magnitudes deeper than the discovery data, allowing us to revisit the properties of these new objects (e.g., distance, structural properties, luminosity measurements, and signs of tidal disturbance). The satellite color-magnitude diagrams show that they are all old (~13.5Gyr) and metal poor ([Fe/H]<~-2.2). Sgr II is particularly interesting, as it sits in an intermediate position between the loci of dwarf galaxies and globular clusters in the size-luminosity plane. The ensemble of its structural parameters is more consistent with a globular cluster classification, indicating that Sgr II is the most extended globular cluster in its luminosity range. The other three satellites land directly on the locus defined by Milky Way ultra-faint dwarf galaxies of similar luminosity. Ret II is the most elongated nearby dwarf galaxy currently known for its luminosity range. Our structural parameters for Phe II and Tuc III suggest that they are both dwarf galaxies. Tuc III is known to be associated with a stellar stream, which is clearly visible in our matched-filter stellar density map. The other satellites do not show any clear evidence of tidal stripping in the form of extensions or distortions. Finally, we also use archival HI data to place limits on the gas content of each object.