Aims. The two stable isotopes of nitrogen, ¹⁴N and ¹⁵N, exhibit a range of abundance ratios both inside and outside the solar system. The elemental ratio in the solar neighborhood is 440. Recent ALMA observations showed HCN/HC¹⁵N ratios from 83 to 156 in six T Tauri and Herbig disks and a CN/C¹⁵N ratio of 323 ± 30 in one T Tauri star. We aim to determine the dominant mechanism responsible for these enhancements of ¹⁵N: low-temperature exchange reactions or isotope-selective photodissociation of N₂. Methods. Using the thermochemical code DALI, we model the nitrogen isotope chemistry in circumstellar disks with a 2D axisymmetric geometry. Our chemical network is the first to include both fractionation mechanisms for nitrogen. The model produces abundance profiles and isotope ratios for several key N-bearing species. We study how these isotope ratios depend on various disk parameters. Results. The formation of CN and HCN is closely coupled to the vibrational excitation of H₂ in the UV-irradiated surface layers of the disk. Isotope fractionation is completely dominated by isotope-selective photodissociation of N₂. The column density ratio of HCN over HC¹⁵N in the disk’s inner 100 au does not depend strongly on the disk mass, the flaring angle or the stellar spectrum, but it is sensitive to the grain size distribution. For larger grains, self-shielding of N₂ becomes more important relative to dust extinction, leading to stronger isotope fractionation. Between disk radii of ~50 and 200 au, the models predict HCN/HC¹⁵N and CN/C¹⁵N abundance ratios consistent with observations of disks and comets. The HCN/HC¹⁵N and CN/C¹⁵N column density ratios in the models are a factor of 2–3 higher than those inferred from the ALMA observations.