This paper presents a domain decomposition-type method for solving real symmetric (Hermitian) eigenvalue problems in which we seek all eigenpairs in an interval [alpha, beta] or a few eigenpairs next to a given real shift zeta. A Newton-based scheme is described whereby the problem is converted to one that deals with the interface nodes of the computational domain. This approach relies on the fact that the inner solves related to each local subdomain are relatively inexpensive. This Newton scheme exploits spectral Schur complements, and these lead to so-called eigenbranches, which are rational functions whose roots are eigenvalues of the original matrix. Theoretical and practical aspects of domain decomposition techniques for computing eigenvalues and eigenvectors are discussed. A parallel implementation is presented and its performance on distributed computing environments is illustrated by means of a few numerical examples.