We present new Chandra X-ray Observatory and Hubble Space Telescope observations of eight optically selected broad-line AGN candidates in nearby dwarf galaxies ( z < 0.055 ) . Including archival Chandra observations of three additional sources , our sample contains all ten galaxies from Reines et al . ( 2013 ) with both broad H \alpha emission and narrow-line AGN ratios ( 6 AGNs , 4 Composites ) , as well as one low-metallicity dwarf galaxy with broad H \alpha and narrow-line ratios characteristic of star formation . All eleven galaxies are detected in X-rays . Nuclear X-ray luminosities range from L _ { 0.5 - 7 keV } \approx 5 \times 10 ^ { 39 } to 1 \times 10 ^ { 42 } erg { s ^ { -1 } } . In all cases except for the star forming galaxy , the nuclear X-ray luminosities are significantly higher than would be expected from X-ray binaries , providing strong confirmation that AGN and composite dwarf galaxies do indeed host actively accreting BHs . Using our estimated BH masses ( which range from \sim 7 \times 10 ^ { 4 } -1 \times 10 ^ { 6 } ~ { } M _ { \odot } ) , we find inferred Eddington fractions ranging from \sim 0.1 - 50 \% , i.e . comparable to massive broad-line quasars at higher redshift . We use the HST imaging to determine the ratio of ultraviolet to X-ray emission for these AGN , finding that they appear to be less X-ray luminous with respect to their UV emission than more massive quasars ( i.e . \alpha _ { OX } values an average of 0.36 lower than expected based on the relation between \alpha _ { OX } and 2500 { \AA } luminosity ) . Finally , we discuss our results in the context of different accretion models onto nuclear BHs .