Using density functional based LDA+U method and linear-response theory, we study the magnetic exchange interactions of the superconductor Nd_1-xSr_xNiO₂. Our calculated nearest-neighbor exchange constant J1=82 meV is large, weakly affected by doping and is only slightly smaller than that found in the sister compound CaCuO₂. We however find that the hole doping significantly enhances the inter-layer exchange coupling as it affects the magnetic moment of the Ni-3d{3z2-r2} orbital. This can be understood in terms of small hybridization of Ni-3d{3z2-r2} within the NiO2 plane which results in a flat band near the Fermi level, and its large overlap along z direction. We also demonstrate that the Nd-5d states appearing at the Fermi level, do not affect the magnetic exchange interactions, and thus may not participate in the superconductivity of this compound. Whereas many previous works emphasize the importance of the Ni-3d{x2-y2} and Nd-5d orbitals, we instead propose that the material can be described by a Ni-3d{x2-y2}/Ni-3d{3z2-r2} two-band model. Its solution is studied here on the level of Dynamical Mean Field Theory and reveals an underlying Mott insulating state which, depending on precise values of the intra-atomic Hunds coupling, produces upon doping competing S=0 and S=1 two-hole states at low energies that lead to very different quasiparticle band structures. We propose that trends upon doping in spin excitational spectrum and quasiparticle density of state can be a way to probe Ni 3d8 configuration.