The connectivity of complex networks and functional implications has been attracting much interest in many physical, biological and social systems. However, the significance of the weight distributions of network links remains largely unknown except for uniformly- or Gaussian-weighted links. Here, we show analytically and numerically, that recurrent neural networks can robustly generate internal noise optimal for spike transmission between neurons with the help of a long-tailed distribution in the weights of recurrent connections. The structure of spontaneous activity in such networks involves weak-dense connections that redistribute excitatory activity over the network as noise sources to optimally enhance the responses of individual neurons to input at sparse-strong connections, thus opening multiple signal transmission pathways. Electrophysiological experiments confirm the importance of a highly broad connectivity spectrum supported by the model. Our results identify a simple network mechanism for internal noise generation by highly inhomogeneous connection strengths supporting both stability and optimal communication.