  The Mg II 2796,2803 doublet is often used to measure interstellar medium
absorption in galaxies, thereby serving as a diagnostic for feedback and
outflows. However, the interpretation of Mg II remains confusing, due to
resonant trapping and re-emission of the photons, analogous to Lyman Alpha.
Therefore, in this paper, we present new MMT Blue Channel Spectrograph
observations of Mg II for a sample of 10 Green Pea galaxies at z~0.2-0.3, where
Lyman Alpha was previously observed with the Cosmic Origins Spectrograph on
Hubble Space Telescope. With strong, (mostly) double-peaked Lyman Alpha
profiles, these galaxies allow us to observe Mg II in the limit of low H I
column density. We find strong Mg II emission and little-to-no absorption. We
use photoionization models to show that nebular Mg II from H II regions is
non-negligible, and the ratios of Mg II/[OIII] 5007 vs. [OIII] 5007/[OII] 3727
form a tight sequence. Using this relation, we predict intrinsic Mg II flux,
and show that Mg II escape fractions range from 0 to 0.9. We find that the Mg
II escape fraction correlates tightly with the Lyman Alpha escape fraction, and
the Mg II line profiles show evidence for broader and more redshifted emission
when the escape fractions are low. These trends are expected if the escape
fractions and velocity profiles of Lyman Alpha and Mg II are shaped by resonant
scattering in the same low column density gas. As a consequence of a close
relation with Lyman Alpha, Mg II may serve as a useful diagnostic in the epoch
of reionization, where Lyman Alpha and Lyman continuum photons are not easily
observed.
