We explore correlations of eigenstates around the many-body localization (MBL) transition in their de-pendence on the energy difference (frequency)ωand disorderW. In addition to the genuine many-bodyproblem, XXZ spin chain in random field, we consider localization on random regular graphs that servesas a toy model of the MBL transition. Both models show a very similar behavior. On the localized side ofthe transition, the eigenstate correlation functionβ(ω) shows a power-law enhancement of correlations withloweringω; the corresponding exponent depends onW. The correlation between adjacent-in-energy eigenstatesexhibits a maximum at the transition pointWc, visualizing the drift ofWcwith increasing system size towardsits thermodynamic-limit value. The correlation functionβ(ω) is related, via Fourier transformation, to theHilbert-space return probability. We discuss measurement of such (and related) eigenstate correlation functionson state-of-the-art quantum computers and simulators.