This paper investigates the problem of linear spatial collaboration for distributed estimation in wireless sensor networks. In this context, the sensors share their local noisy (and potentially spatially correlated) observations with each other through error-free, low cost links based on a pattern defined by an adjacency matrix. Each sensor connected to a central entity, known as the fusion center (FC), forms a linear combination of the observations to which it has access and sends the resulting signal to the FC through an orthogonal fading channel. The FC combines these received signals to find the best linear unbiased estimator of the vector of unknown signals observed by individual sensors. The main novelty of this paper is the derivation of an optimal power-allocation scheme in which the coefficients used to form linear combinations of noisy observations at the sensors connected to the FC are optimized. Through this optimization, the total estimation distortion at the FC is minimized, given a constraint on the maximum cumulative transmit power in the entire network. Numerical results show that even with a moderate connectivity across the network, spatial collaboration among sensors significantly reduces the estimation distortion at the FC.