Using density-functional theory (DFT) calculations we probe the spin polarization of functionalized two-dimensional (2D) phthalo-carbonitrides (pc-C₃N₂), i.e., 2D polymers of tetra-cyanoethylene. The attachment of various metal atoms at the cores of these covalent organic frameworks gives rise to diverse electronic and magnetic configurations. In particular, Fe- and Co-functionalized systems are ferromagnetic (FM) metals, or anti-ferromagnetic (AFM) semiconductors within the DFT+U approach. Cu- and Cr-laden systems are AFM semiconductors, while FM and AFM spin structures of pc-C₃N₂ sheets decorated with Mn atoms are almost degenerate in terms of energy. The DFT (DFT+U) results find the FM configuration of Mn-functionalized pc-C₃N₂ to be half-metallic (semi-metallic). Based on the above, metal-functionalized pc-C₃N₂ could offer new functional components in 2D-based electronic and spintronic devices.