In this paper, we consider the issue of distributed computation of tensor decompositions. A central unit observing a global data tensor assigns different data sub-tensors to several computing nodes grouped into clusters. The goal is to distribute the computation of a tensor decomposition across the different computing nodes of the network, which is particularly useful when dealing with large-scale data tensors. However, this is only possible when the data sub-tensors assigned to each computing node in a cluster satisfies minimum conditions for uniqueness. By allowing collaboration between computing nodes in a cluster, we show that average consensus based estimation is useful to yield unique estimates of the factor matrices of each data sub-tensor. Moreover, an essentially unique reconstruction of the global factor matrices at the central unit is possible by allowing the subtensors assigned to different clusters to overlap in one mode. The proposed approach may be useful to a number of distributed tensor-based estimation problems in signal processing.