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We report the bivariate HI- and H$_2$-stellar mass distributions of local galaxies in addition of an inventory of galaxy mass functions, MFs, for HI, H$_2$, cold gas, and baryonic mass, separately into early- and late-type galaxies. The MFs are determined using the HI and H$_2$ conditional distributions and the galaxy stellar mass function, GSMF. For the conditional distributions we use the compilation presented in Calette et al. 2018. For determining the GSMF from $M_{\ast}\sim3\times10^{7}$ to $3\times10^{12}$ $M_{\odot}$, we combine two spectroscopic samples from the SDSS at the redshift range $0.0033<z<0.2$. We find that the low-mass end slope of the GSMF, after correcting from surface brightness incompleteness, is $α\approx-1.4$, consistent with previous determinations. The obtained HI MFs agree with radio blind surveys. Similarly, the H$_2$ MFs are consistent with CO follow-up optically-selected samples. We estimate the impact of systematics due to mass-to-light ratios and find that our MFs are robust against systematic errors. We deconvolve our MFs from random errors to obtain the intrinsic MFs. Using the MFs, we calculate cosmic density parameters of all the baryonic components. Baryons locked inside galaxies represent 5.4% of the universal baryon content, while $\sim96$% of the HI and H$_2$ mass inside galaxies reside in late-type morphologies. Our results imply cosmic depletion times of H$_2$ and total neutral H in late-type galaxies of $\sim 1.3$ and 7.2 Gyr, respectively, which shows that late type galaxies are on average inefficient in converting H$_2$ into stars and in transforming HI gas into H$_2$. Our results provide a fully self-consistent empirical description of galaxy demographics in terms of the bivariate gas--stellar mass distribution and their projections, the MFs. This description is ideal to compare and/or to constrain galaxy formation models.