We address the consensus-based distributed linear filtering problem, where a discrete time, linear stochastic process is observed by a network of sensors. We assume that the consensus weights are known and we first provide sufficient conditions under which the stochastic process is detectable, i.e. for a specific choice of consensus weights there exists a set of filtering gains such that the dynamics of the estimation errors (without noise) is asymptotically stable. Next, we provide a distributed sub-optimal filtering scheme based on optimizing an upper bound on a quadratic filtering cost. In the stationary case, we provide sufficient conditions under which this scheme converges; conditions expressed in terms of the convergence properties of a set of coupled Riccati equations. We continue with presenting a connection between the consensus-based distributed linear filter and the optimal linear filter of a Markovian jump linear system, appropriately defined. More specifically, we show that if the Markovian jump linear system is (mean square) detectable, then the stochastic process is detectable under the consensus-based distributed linear filtering scheme. We also show that the state estimate given by the optimal linear filter of a Markovian jump linear system appropriately defined can be seen as an approximation of the optimal average state estimate obtained using the consensus-based linear filtering scheme.