In this paper we consider the problem of recovering random graph signals by using graph signal processing (GSP) tools. We focus on partially-known linear settings, where one has access to data in order to cope with the missing domain knowledge in designing a graph filter for signal recovery. In this work, we formulate two main approaches for leveraging both the available domain knowledge and data for such graph filter design: 1) the GSP-generative approach, where data is used to fit the underlying linear model that determines the graph filter; and 2) the GSP-discriminative approach, where data is used to directly learn the graph filter for graph signal recovery, bypassing the need to estimate the underlying model. Then, we compare qualitatively and quantitatively these two approaches of graph filter design. Our results provide an understanding with regard to which approach is preferable in which regime. In particular, it is shown that GSP-discriminative learning reliably copes with mismatches in the available domain knowledge, since it bypasses the need to fit the underlying model. On the other hand, the model awareness of the GSP-generative approach results in its achieving a lower mean-squared error (MSE) when data is scarce. In the asymptotic region where the number of training data points approaches infinity, both approaches achieve the oracle minimum MSE estimator under the considered setting.