There is currently much interest in the recycling of entangled systems, for use in quantum information protocols by sequential observers. In this work, we study the sequential generation of Bell nonlocality via recycling one or both components of two-qubit states. We first give a description of two-valued qubit measurements in terms of measurement bias, strength, and reversibility, and derive useful tradeoff relations between them. Then, we derive one-sided monogamy relations that support the recent Conjecture in [S. Cheng {it et al.}, arXiv:2102.11574], that if the first pair of observers violate Bell nonlocality then a subsequent independent pair cannot. We also answer a question raised in [P. J. Brown and R. Colbeck, Phys. Rev. Lett. textbf{125}, 090401 (2020)], by showing that the conditions given therein for the recycling of one qubit by an arbitrarily large number of observers are sufficient but not necessary. Finally, we find that it is possible to share Bell nonlocality between multiple pairs of independent observers on both sides, if sufficiently many pairs of qubits are shared. Our results are based on a formalism that is applicable to more general problems in recycling entanglement, and hence is expected to aid progress in this field.
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