We introduce distributed NLI, a new NLU task with a goal to predict the distribution of human judgements for natural language inference. We show that models can capture human judgement distribution by applying additional distribution estimation methods, namely, Monte Carlo (MC) Dropout, Deep Ensemble, Re-Calibration, and Distribution Distillation. All four of these methods substantially outperform the softmax baseline. We show that MC Dropout is able to achieve decent performance without any distribution annotations while Re-Calibration can further give substantial improvements when extra distribution annotations are provided, suggesting the value of multiple annotations for the example in modeling the distribution of human judgements. Moreover, MC Dropout and Re-Calibration can achieve decent transfer performance on out-of-domain data. Despite these improvements, the best results are still far below estimated human upper-bound, indicating that the task of predicting the distribution of human judgements is still an open, challenging problem with large room for future improvements. We showcase the common errors for MC Dropout and Re-Calibration. Finally, we give guidelines on the usage of these methods with different levels of data availability and encourage future work on modeling the human opinion distribution for language reasoning.