A distributed least-squares estimation strategy is developed by appealing to collaboration techniques that exploit the space-time structure of the data, achieving an exact recursive solution that is fully distributed. Each node is allowed to communicate with its immediate neighbor in order to exploit the spatial dimension, while it evolves locally to account for the time dimension as well. In applications where communication and energy resources are scarce, an approximate RLS scheme that is also fully distributed is proposed in order to decrease the communication burden necessary to implement distributed collaborative solution. The performance of the resulting algorithm tends to its exact counterpart in the mean-square sense as the forgetting factor lambda tends to unity. A spatial-temporal energy conservation argument is used to evaluate the steady-state performance of the individual nodes across the adaptive distributed network for the low communications RLS implementation. Computer simulations illustrate the results.