Merger simulations predict that tidally induced gas inflows can trigger kpc-scale dual active galactic nuclei (dAGN) in heavily obscured environments. Previously with the Very Large Array, we have confirmed four dAGN with redshifts between $0.04 < z < 0.22$ and projected separations between 4.3 and 9.2 kpc in the SDSS Stripe 82 field. Here, we present $Chandra$ X-ray observations that spatially resolve these dAGN and compare their multi-wavelength properties to those of single AGN from the literature. We detect X-ray emission from six of the individual merger components and obtain upper limits for the remaining two. Combined with previous radio and optical observations, we find that our dAGN have properties similar to nearby low-luminosity AGN, and they agree well with the black hole fundamental plane relation. There are three AGN-dominated X-ray sources, whose X-ray hardness-ratio derived column densities show that two are unobscured and one is obscured. The low obscured fraction suggests these dAGN are no more obscured than single AGN, in contrast to the predictions from simulations. These three sources show an apparent X-ray deficit compared to their mid-infrared continuum and optical [OIII] line luminosities, suggesting higher levels of obscuration, in tension with the hardness-ratio derived column densities. Enhanced mid-infrared and [OIII] luminosities from star formation may explain this deficit. There is ambiguity in the level of obscuration for the remaining five components since their hardness ratios may be affected by non-nuclear X-ray emissions, or are undetected altogether. They require further observations to be fully characterized.
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