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In this paper, we describe a general algorithmic framework for solving linear signal or feature fusion optimization problems in a distributed setting, for example in a wireless sensor network (WSN). These problems require linearly combining the observed signals (or features thereof) collected at the various sensor nodes to satisfy a pre-defined optimization criterion. The framework covers several classical spatial filtering problems, including minimum variance beamformers, multi-channel Wiener filters, principal component analysis, canonical correlation analysis, (generalized) eigenvalue problems, etc. The proposed distributed adaptive signal fusion (DASF) algorithm is an iterative method that solves these types of problems by allowing each node to share a linearly compressed version of the local sensor signal observations with its neighbors to reduce the energy and bandwidth requirements of the network. We first discuss the case of fully-connected networks and then extend the analysis to more general network topologies. The general DASF algorithm is shown to have several existing distributed algorithms from the literature as a special case, while at the same time allowing to solve new distributed problems as well with guaranteed convergence and optimality. This paper focuses on the algorithm derivation of the DASF framework along with simulations demonstrating its performance. A technical analysis along with convergence conditions and proofs are provided in a companion paper.