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High temperature superconductivity remains challenging more than thirty years after its initial discovery. One of the most captivating ongoing debates is about the mechanism and the normal state above the critical temperature. Various experiments imply paired charge carriers persisting above the transition temperature T_c, but without direct signatures of pairing. Shot noise, the intrinsic current fluctuations that result from the discreteness of current-carrying excitations, probes the magnitude of the charge of carriers. We report measurements on the shot noise of the tunneling current in c-axis epitaxial structure of La_2-xSr_xCuO₄/La₂CuO₄/La_2-xSr_xCuO₄ for several doping levels. At temperatures far above T_c, the shot noise agrees quantitatively with independent tunneling of carriers with charge magnitude e. Approaching but still above T_c, the noise exceeds the expectations of single charge tunneling, indicating paring of carriers. Well below T_c the noise is greatly enhanced within the bias region of the superconducting gap, consistent with multiple Andreev reflection processes.