We use spectral stacking to measure the contribution of galaxies of different masses and in different hierarchies to the cosmic atomic hydrogen (HI) mass density in the local Universe. Our sample includes 1793 galaxies at z <0.11 observed with the Westerbork Synthesis Radio Telescope, for which Sloan Digital Sky Survey spectroscopy and hierarchy information are also available. We find a cosmic III mass density of Omega(HI) = (3.99 +/- 0.54) x 10(-4) h(70)(-1) at <z > = 0.065. For the central and satellite galaxies, we obtain Omega(HI) of (3.51 +/- 0.49) x 10(-4) h(70)(-1) and (0.90 +/- 0.16) x 10(-4) h(70)(-1), respectively. We show that galaxies above and below stellar masses of similar to 10(9.3) M-circle dot contribute in roughly equal measure to the global value of Omega(HI). While consistent with estimates based on targeted III surveys, our results are in tension with previous theoretical work. We show that these differences are, at least partly, due to the empirical recipe used to set the partition between atomic and molecular hydrogen in semi-analytical models. Moreover, comparing our measurements with the cosmological semi-analytic models of galaxy formation SHARK and GALFORM reveals gradual stripping of gas via ram pressure works better to fully reproduce the properties of satellite galaxies in our sample than strangulation. Our findings highlight the power of this approach in constraining theoretical models and confirm the non-negligible contribution of massive galaxies to the HI mass budget of the local Universe.