Nested named entity recognition (nested NER) is a fundamental task in natural language processing. Various span-based methods have been proposed to detect nested entities with span representations. However, span-based methods do not consider the relationship between a span and other entities or phrases, which is helpful in the NER task. Besides, span-based methods have trouble predicting long entities due to limited span enumeration length. To mitigate these issues, we present the Propose-and-Refine Network (PnRNet), a two-stage set prediction network for nested NER. In the propose stage, we use a span-based predictor to generate some coarse entity predictions as entity proposals. In the refine stage, proposals interact with each other, and richer contextual information is incorporated into the proposal representations. The refined proposal representations are used to re-predict entity boundaries and classes. In this way, errors in coarse proposals can be eliminated, and the boundary prediction is no longer constrained by the span enumeration length limitation. Additionally, we build multi-scale sentence representations, which better model the hierarchical structure of sentences and provide richer contextual information than token-level representations. Experiments show that PnRNet achieves state-of-the-art performance on four nested NER datasets and one flat NER dataset.