The shared information between two parts of a systems has been studied extensively in the context of quantum spin chains, where it was found to exhibit universal features at quantum critical points. It has been suggested recently that similar universal properties can be found in classical spin chains. We study this issue in a classical one-dimensional system with a general type of infinite-range and short-range interactions, where shared-information estimators can be computed analytically. The bipartite mutual information is found to diverge at most logarithmically with the system-size, despite the existence of long-range interactions. The coefficient in front of the logarithm is independent of any details of the model, and thus assumed to characterize the mean-field universality class. In general, such coefficients may present a new way for characterizing universality classes of classical spins chains, both in and out of equilibrium.