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We present families of beating dark-dark soiitons in spin-orbit (SO) -coupled Bose-Einstein condensates. These families consist of soiitons residing simultaneously in the two bands of the energy spectrum. The soliton components are characterized by two different spatial and temporal scales, which are identified by a multiscale expansion method. The soiitons are "beating" ones, as they perform density oscillations. The characteristic frequency of the latter is relevant to Zitterbewegung (ZB) oscillations, which were recently observed in experiments with SO-coupled condensates [C. Qu et al., Phys. Rev. A 88, 021604(R) (2013); L. J. LeBlanc etal.,New J. Phys. 15, 073011 (2013)]. We find that spin oscillations may occur, depending on the parity of each soliton branch, which consequently lead to ZB oscillations of the beating dark soiitons. Analytical results are corroborated by numerical simulations, illustrating the robustness of the soiitons. [ABSTRACT FROM AUTHOR]