To advance quantum information science, physical systems are sought that meet the stringent requirements for creating and preserving quantum entanglement. In atomic physics, robust two-qubit entanglement is typically achieved by strong, long-range interactions in the form of either Coulomb interactions between ions or dipolar interactions between Rydberg atoms(1-4). Although such interactions allow fast quantum gates, the interacting atoms must overcome the associated coupling to the environment and crosstalk among qubits(5-)8. Local interactions, such as those requiring substantial wavefunction overlap, can alleviate these detrimental effects; however, such interactions present a new challenge: to distribute entanglement, qubits must be transported, merged for interaction, and then isolated for storage and subsequent operations. Here we show how, using a mobile optical tweezer, it is possible to prepare and locally entangle two ultracold neutral atoms, and then separate them while preserving their entanglement(9-11). Ground-state neutral atom experiments have measured dynamics consistent with spin entanglement(10,12,13), and have detected entanglement with macroscopic observables(14,15); we are now able to demonstrate position-resolved two-particle coherence via application of a local gradient and parity measurements . This new entanglement-verification protocol could be applied to arbitrary spin-entangled states of spatially separated atoms(16,17). The local entangling operation is achieved via spin-exchange interactions(9-11), and quantum tunnelling is used to combine and separate atoms. These techniques provide a framework for dynamically entangling remote qubits via local operations within a large-scale quantum register.