Execution of functional programs on distributed-memory multiprocessors gives rise to the problem of evaluating expressions that are shared between several Processing Elements (PEs). One of the main difficulties of solving this problem is that, for a given shared expression, it is not known in advance whether realizing the sharing is more cost effective than duplicating its evaluation. Realizing the sharing requires coordination between the sharing PEs to ensure that the shared expression is evaluated only once. This coordination involves relatively high communication costs, and is therefore only worthwhile when the shared expressions require much computation time to evaluate. In contrast, when the shared expression is not computation intensive, it is more cost effective to duplicate the evaluation, and thus avoid the communication overhead costs. This dilemma of deciding whether to duplicate the work or to realize the sharing stems from the unknown computation time that is required to evaluate a shared expression. This computation time is difficult to estimate due to unknown run-time evolution of loops and recursion that may be part of the expression. This paper presents an on-line (run-time) algorithm that decides which of the expressions that are shared between several PEs should be evaluated only once, and which expressions should be evaluated locally by each sharing PE. By applying competitive considerations, the algorithm manages to exploit sharing of computation-intensive expressions, while it duplicates the evaluation of expressions that require little time to compute. The algorithm accomplishes this goal even though it has no a priori knowledge of the amount of computation that is required to evaluate the shared expression. We show that this algorithm is competitive with a hypothetical optimal off-line algorithm, which does have such knowledge, and we prove that the algorithm is deadlock free. Furthermore, this algorithm does not require any programmer intervention, it has low overhead, and it is designed to run on a wide variety of distributed systems.