There is a dual description of $\overline d$ which goes back to Banach and Følner (in the general countable case), see [Theorem 6 of Granirer][1]. For locally compact groups it should be more or less the same if you deal with the means defined on $L^\infty$ with respect to the Haar measure.


  [1]: https://mathscinet.ams.org/mathscinet-getitem?mr=283171