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The continued desire for X-ray pixel detectors with higher frame rates will stress the ability of application-specific integrated circuit (ASIC) designers to provide sufficient off-chip bandwidth to reach continuous frame rates in the 1 MHz regime. To move from the current 10 kHz to the 1 MHz frame rate regime, ASIC designers will continue to pack as many power-hungry high-speed transceivers at the periphery of the ASIC as possible. In this paper, however, we present new strategies to make the most efficient use of the off-chip bandwidth by utilizing data compression schemes for X-ray photon-counting and charge-integrating pixel detectors. In particular, we describe a novel in-pixel compression scheme that converts from analog to digital converter units to encoded photon counts near the photon Poisson noise level and achieves a compression ratio of $>\!1.5\times$ independent of the dataset. In addition, we describe a simple yet efficient zero-suppression compression scheme called ``zeromask'' (ZM) located at the ASIC's edge before streaming data off the ASIC chip. ZM achieves average compression ratios of $>\!4\times$, $>\!7\times$, and $>\!8\times$ for high-energy X-ray diffraction, ptychography, and X-ray photon correlation spectroscopy datasets, respectively. We present the conceptual designs, register-transfer level block diagrams, and the physical ASIC implementation of these compression schemes in 65-nm CMOS. When combined, these two digital compression schemes could increase the effective off-chip bandwidth by a factor of 6-12$\times$.