We have examined a subset of 18 active galactic nuclei (AGNs) drawn from a sample of 81 galaxies that possess double-peaked narrow optical emission line spectra in the Sloan Digital Sky Survey, have 2 optical AGN emission components separated by >0.″2, and are detected in the Faint Images of the Radio Sky at Twenty-centimeters survey. Without follow-up observations, the sources of the double-peaked narrow emission lines are uncertain, and may be produced by kiloparsec-scale separation dual active supermassive black holes, AGN outflows, or disk rotation. In this work, we propose a new methodology to characterize double-peaked narrow emission line galaxies based on optical long-slit spectroscopy and high-resolution multi-band Very Large Array observations. The nature of the radio emission in the sample galaxies is varied. Of the 18 galaxies, we detect 2 compact flat-spectrum radio cores with projected spatial separations on the sky between 0.6 and 1.6 kpc in 3 galaxies: J1023+3243, J1158+3231, and J1623+0808. The two radio sources are spatially coincident with the two optical components of ionized gas with AGN-like line ratios, which confirms the presence of dual AGNs in these three galaxies. Dual AGNs account for only ∼15% (3/18) of the double-peaked AGNs in our sample. Gas kinematics produce ∼75% (13/18) of the double-peaked narrow emission lines, distributed in the following way: seven AGN wind-driven outflows, five radio-jet driven outflows, and one rotating narrow-line region. The remaining 10% (2/18) are ambiguous cases. Our method demonstrates the power of spatially resolved spectroscopy and high-resolution radio observations for the identification of AGN outflows and AGN pairs with angular separations as small as 0.″18.