We present a new second-order unstaggered semi-discrete path-conservative central-upwind (PCCU) scheme for ideal and shallow water magnetohydrodynamics (MHD) equations. The new scheme possesses several important properties: it locally preserves the divergence-free constraint, it does not rely on any (approximate) Riemann problem solver, and it robustly produces high-resolution and non-oscillatory results. The derivation of the scheme is based on the Godunov-Powell neoconservative modifications of the studied MHD systems. The local divergence-free property is enforced by augmenting the modified systems with the evolution equations for the corresponding derivatives of the magnetic field components. These derivatives are then used to design a special piecewise linear reconstruction of the magnetic field, which guarantees a non-oscillatory nature of the resulting scheme. In addition, the proposed PCCU discretization accounts for the jump of the nonconservative product terms across cell interfaces, thereby ensuring stability. The new scheme has been tested on benchmarks for ideal and shallow water MHD systems. The obtained numerical results illustrate the performance of the new scheme, its robustness, and its ability not only to achieve high resolution but also to preserve the positivity of computed quantities such as density, pressure, and water depth, as well as steady-state solutions.