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Hubbard-corrected density-functional theory has proven to be successful inaddressing self-interaction errors in 3D magnetic materials. However, theeffectiveness of this approach for 2D magnetic materials has not beenextensively explored. Here, we use PBEsol+$\textit{U}$ and its extensionsPBEsol+$\textit{U}$+$\textit{V}$ to investigate the electronic, structural, andvibrational properties of 2D antiferromagnetic FePS$_3$ and ferromagneticCrI$_3$, and compare the monolayers with their bulk counterparts. Hubbardparameters (on-site $\textit{U}$ and inter-site $\textit{V}$) are computedself-consistently using density-functional perturbation theory, thus avoidingany empirical assumptions. We show that for FePS$_3$ the Hubbard correctionsare crucial in obtaining the experimentally observed insulating state with thecorrect crystal symmetry, providing also vibrational frequencies in goodagreement with Raman experiments. For ferromagnetic CrI$_3$, we discuss how astraightforward application of Hubbard corrections worsens the results andintroduces a spurious separation between spin-majority and minority conductionbands. Promoting the Hubbard $\textit{U}$ to be a spin-resolved parameter -that is, applying different (first-principles) values to the spin-up andspin-down manifolds - recovers a more physical picture of the electronic bandsand delivers the best comparison with experiments.

On-site and inter-site Hubbard corrections in magnetic monolayers: The case of FePS$_3$ and CrI$_3$