ABSTRACT Owing to their quiet evolutionary histories, nearby dwarf galaxies (stellar masses $M_⋆ \lesssim 3 \times 10^9 \, \mathrm{M}_⊙$) have the potential to teach us about the mechanism(s) that ‘seeded’ the growth of supermassive black holes, and also how the first stellar mass black holes formed and interacted with their environments. Here, we present high spatial resolution observations of three dwarf galaxies in the X-ray (Chandra), the optical/near-infrared (Hubble Space Telescope), and the radio (Karl G. Jansky Very Large Array). These three galaxies were previously identified as hosting candidate active galactic nuclei on the basis of lower resolution X-ray imaging. With our new observations, we find that X-ray sources in two galaxies (SDSS J121326.01+543631.6 and SDSS J122111.29+173819.1) are off-nuclear and lack corresponding radio emission, implying they are likely luminous X-ray binaries. The third galaxy (Mrk 1434) contains two X-ray sources (each with LX ≈ 1040 erg s−1) separated by 2.8 arcsec, has a low metallicity [12 + log(O/H) = 7.8], and emits nebular He ii λ4686 line emission. The northern source has spatially coincident point-like radio emission at 9.0 GHz and extended radio emission at 5.5 GHz. We discuss X-ray binary interpretations (where an ultraluminous X-ray source blows a ‘radio bubble’) and active galactic nucleus interpretations (where an $≈ 4\times 10^5 \, \mathrm{M}_⊙$ black hole launches a jet). In either case, we find that the He ii emission cannot be photoionized by the X-ray source, unless the source was ≈30–90 times more luminous several hundred years ago.