While distant supervision has been extensively explored and exploited in NLP tasks like named entity recognition, a major obstacle stems from the inevitable noisy distant labels tagged unsupervisedly. A few past works approach this problem by adopting a self-training framework with a sample-selection mechanism. In this work, we innovatively identify two types of biases that were omitted by prior work, and these biases lead to inferior performance of the distant-supervised NER setup. First, we characterize the noise concealed in the distant labels as highly structural rather than fully randomized. Second, the self-training framework would ubiquitously introduce an inherent bias that causes erroneous behavior in both sample selection and eventually prediction. To cope with these problems, we propose a novel self-training framework, dubbed DesERT. This framework augments the conventional NER predicative pathway to a dual form that effectively adapts the sample-selection process to conform to its innate distributional-bias structure. The other crucial component of DesERT composes a debiased module aiming to enhance the token representations, hence the quality of the pseudo-labels. Extensive experiments are conducted to validate the DesERT. The results show that our framework establishes a new state-of-art performance, it achieves a +2.22% average F1 score improvement on five standardized benchmarking datasets. Lastly, DesERT demonstrates its effectiveness under a new DSNER benchmark where additional distant supervision comes from the ChatGPT model.