In distributed training, the communication cost due to the transmission of gradients or the parameters of the deep model is a major bottleneck in scaling up the number of processing nodes. To address this issue, we propose \emph{dithered quantization} for the transmission of the stochastic gradients and show that training with \emph{Dithered Quantized Stochastic Gradients (DQSG)} is similar to the training with unquantized SGs perturbed by an independent bounded uniform noise, in contrast to the other quantization methods where the perturbation depends on the gradients and hence, complicating the convergence analysis. We study the convergence of training algorithms using DQSG and the trade off between the number of quantization levels and the training time. Next, we observe that there is a correlation among the SGs computed by workers that can be utilized to further reduce the communication overhead without any performance loss. Hence, we develop a simple yet effective quantization scheme, nested dithered quantized SG (NDQSG), that can reduce the communication significantly \emph{without requiring the workers communicating extra information to each other}. We prove that although NDQSG requires significantly less bits, it can achieve the same quantization variance bound as DQSG. Our simulation results confirm the effectiveness of training using DQSG and NDQSG in reducing the communication bits or the convergence time compared to the existing methods without sacrificing the accuracy of the trained model.