In multi-task learning, we often encounter the case when the presence of labels across samples exhibits irregular patterns: samples can be fully labeled, partially labeled or unlabeled. Taking drug analysis as an example, multiple toxicity properties of a drug molecule may not be concurrently available due to experimental limitations. It triggers a demand for a new training and inference mechanism that could accommodate irregularly present labels and maximize the utility of any available label information. In this work, we focus on the two-label learning task, and propose a novel training and inference framework, Dual-Label Learning (DLL). The DLL framework formulates the problem into a dual-function system, in which the two functions should simultaneously satisfy standard supervision, structural duality and probabilistic duality. DLL features a dual-tower model architecture that explicitly captures the information exchange between labels, aimed at maximizing the utility of partially available labels in understanding label correlation. During training, label imputation for missing labels is conducted as part of the forward propagation process, while during inference, labels are regarded as unknowns of a bivariate system of equations and are solved jointly. Theoretical analysis guarantees the feasibility of DLL, and extensive experiments are conducted to verify that by explicitly modeling label correlation and maximizing the utility of available labels, our method makes consistently better predictions than baseline approaches by up to a 10% gain in F1-score or MAPE. Remarkably, our method provided with data at a label missing rate as high as 60% can achieve similar or even better results than baseline approaches at a label missing rate of only 10%.