  Ly$\alpha$ emission is a standard tracer of starburst galaxies at high
redshift. However, a number of local Ly$\alpha$ emitters (LAEs) are X-ray
sources, suggesting a possible origin of Ly$\alpha$ photons other than young,
hot stars, and which may be active at much later ages relative to the parent
starburst. Resolved, nearby LAEs offer the opportunity to discriminate between
diffuse X-ray emission arising from supernova-heated gas, high-mass X-ray
binaries (HMXBs), or low-luminosity active galactic nuclei (LLAGN). We examine
archival X-ray imaging from Chandra and XMM-Newton for 11 galaxies with
spatially resolved Ly$\alpha$ imaging to determine the luminosity, morphology,
and spectral hardness of the X-ray sources. The data are consistent with 9 of
the 12, bright Ly$\alpha$ sources being driven by luminous, $10^{40}$ erg
s$^{-1}$ X-ray sources. Half of the 8 Chandra sources are unresolved. The data
suggest that nuclear activity, whether from LLAGN or nuclear starbursts, may
play an important role in Ly$\alpha$ emission. Our results also suggest a
significant link between Ly$\alpha$ emission and HMXBs, ULXs, and/or LLAGN,
which would imply that Ly$\alpha$ may be generated over timescales 1 - 2 orders
of magnitude longer than produced by photoionization from OB stars. This
highlights a critical need to quantify the relative contributions of different
sources across cosmic time, to interpret Ly$\alpha$ observations and the
resulting properties of distant galaxies.
