We study collective "free-space" radiation properties of two distant single-layer arrays of quantum emitters as two-level atoms. We show that this system can support a long-lived Bell superposition state of atomic excitations exhibiting strong subradiance, which corresponds to a nonlocal excitation of the two arrays. We describe the preparation of these states and their application in quantum information as a resource of nonlocal entanglement, including deterministic quantum state transfer with high fidelity between the arrays representing quantum memories. We discuss experimental realizations using cold atoms in optical trap arrays with subwavelength spacing, and analyze the role of imperfections.