This paper considers distributed detection over a noisy network, in which each connected sensor pair can communicate over an additive noise channel. With non-identically distributed generic sensor observations, a mixed time scale recursive algorithm for binary hypothesis testing over such networks is proposed. Under some mild assumptions on network connectivity and global detectability (the positivity of the global or centralized Kullback-Liebler divergence), this algorithm yields asymptotically zero probabilities of Type-I and Type-II errors (henceforth referred to as probabilities of error). When sensor observations are identically distributed, a simplified single time scale version of the proposed algorithm is shown to achieve asymptotically zero probabilities of error. Convergence rate guarantees in terms of asymptotic normality of certain scaled decision variables are provided for this simplified procedure. As an example, a practical Gaussian sensor network is considered, for which the error decay exponents are explicitly characterized in terms of the network and noise parameters.