Knowledge of the shape of the mass spectrum of compact objects can be used to help break the degeneracy between the mass and redshift of the gravitational wave (GW) sources, and thus can be used to infer cosmological parameters in the absence of redshift measurements obtained from electromagnetic observations. In this paper, we study extensively different aspects of this approach, including its computational limits and achievable accuracy. We focus on ground-based detectors with current and future sensitivities, we first perform the analysis of an extensive set of simulated data with a hierarchical Bayesian scheme inferring population and cosmological parameters. We consider a population model (power-law plus Gaussian) which exhibits characteristic scales (extremes of the mass spectrum, presence of an accumulation point) that allows an indirect estimate of the source redshift. Our analysis of this catalog highlights and quantifies the tight interplay between source population and cosmological parameters, as well as the influence of initial assumptions (whether formulated on the source or cosmological parameters). We then validate our results by an end-to-end analysis using simulated GW data and posterior samples generated from Bayesian samplers used for GW parameter estimation, thus mirroring the analysis chain used for observational data for the first time in literature. Our results then lead us to re-examine the estimation of $H_0$ obtained with GWTC-1, and we show explicitly how population assumptions impact the final $H_0$ result. Our results underline the importance of inferring population and cosmological parameters jointly (and not separately as is often assumed). The only exception, as we discuss, is if an electromagnetic counterpart was to be observed for all the BBH events: then the population assumptions have less impact on the estimation of cosmological parameters.
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