'Blinking', or 'fluorescence intermittency', refers to a random switching between 'ON' (bright) and 'OFF' (dark) states of an emitter; it has been studied widely in zero-dimensional quantum dots(1) and molecules(2,3), and scarcely in one-dimensional systems(4,5). A generally accepted mechanism for blinking in quantum dots involves random switching between neutral and charged states(6,7) (or is accompanied by fluctuations in charge-carrier traps(8)), which substantially alters the dynamics of radiative and non-radiative decay. Here, we uncover a new type of blinking effect in vertically stacked, two-dimensional semiconductor heterostructures(9), which consist of two distinct monolayers of transition metal dichalcogenides (TMDs) that are weakly coupled by van der Waals forces. Unlike zero-dimensional or one-dimensional systems, two-dimensional TMD heterostructures show a correlated blinking effect, comprising randomly switching bright, neutral and dark states. Fluorescence cross-correlation spectroscopy analyses show that a bright state occurring in one monolayer will simultaneously lead to a dark state in the other monolayer, owing to an intermittent interlayer carrier-transfer process. Our findings suggest that bilayer van der Waals heterostructures provide unique platforms for the study of charge-transfer dynamics and non-equilibrium-state physics, and could see application as correlated light emitters in quantum technology.