We present and analyze two algorithms - termed distributed stochastic approximation mirror descent (D-SAMD) and accelerated distributed stochastic approximation mirror descent (AD-SAMD) - for distributed, stochastic optimization from high-rate data streams over rate-limited networks. Devices contend with fast streaming rates by mini-batching samples in the data stream, and they collaborate via distributed consensus to compute variance-reduced averages of distributed subgradients. This induces a trade-off: Mini-batching slows down the effective streaming rate, but may also slow down convergence. We present two theoretical contributions that characterize this trade-off: (i) bounds on the convergence rates of D-SAMD and AD-SAMD, and (ii) sufficient conditions for order-optimum convergence of D-SAMD and AD-SAMD, in terms of the network size/topology and the ratio of the data streaming and communication rates. We find that AD-SAMD achieves order-optimum convergence in a larger regime than D-SAMD. We demonstrate the effectiveness of the proposed algorithms using numerical experiments.