Context. Molecules containing two or more hydrogen or deuterium atoms have different nuclear spin states which behave as separate chemical species. The relative abundances of these species can give clues to their origin. Formation on grains is believed to yield statistical spin ratios whereas gas-phase reactions are predicted to result in clear deviations from them. This is also true for ammonia and its deuterated forms NH2D, NHD2, and ND3. Aims. Here we aim to determine the ortho/para ratios of NH2D and NHD2 in dense, starless cores, where their formation is supposed to be dominated by gas-phase reactions. Methods. The Large APEX sub-Millimeter Array (LAsMA) multibeam receiver of the Atacama Pathfinder EXperiment (APEX) telescope was used to observe the prestellar cores H-MM1 and Oph D in Ophiuchus in the ground-state lines of ortho and para NH2D and NHD2. The fractional abundances of these molecules were derived employing three-dimensional radiative transfer modelling, using different assumptions about the abundance profiles as functions of density. We also ran gas-grain chemistry models with different scenarios concerning proton or deuteron exchanges and chemical desorption from grains to find out if one of these models can reproduce the observed spin ratios. Results. The observationally deduced ortho/para ratios of NH2D and NHD2 are in both cores within 10% of their statistical values 3 and 2, respectively, and taking 3 s limits, deviations from these of about 20% are allowed. Of the chemistry models tested here, the model that assumes proton hop (as opposed to full scrambling) in reactions contributing to ammonia formation, and a constant efficiency of chemical desorption, comes nearest to the observed abundances and spin ratios. Conclusions. The nuclear spin ratios derived here are in contrast with spin-state chemistry models that assume full scrambling in proton donation and hydrogen abstraction reactions leading to deuterated ammonia. The efficiency of chemical desorption strongly influences the predicted abundances of NH3, NH2D, and NHD2, but has a lesser effect on their ortho/para ratios. For these the proton exchange scenario in the gas is decisive. We suggest that this is because of rapid re-processing of ammonia and related cations by gas-phase ion-molecule reactions. © The Authors 2024.