  Physical principles constrain the way nonlocal correlations can be
distributed among distant parties. These constraints are usually expressed by
monogamy relations that bound the amount of Bell inequality violation observed
among a set of parties by the violation observed by a different set of parties.
We prove here that much stronger monogamy relations are possible for
nonsignalling correlations by showing how nonlocal correlations among a set of
parties limit any form of correlations, not necessarily nonlocal, shared among
other parties. In particular, we provide tight bounds between the violation of
a family of Bell inequalities among an arbitrary number of parties and the
knowledge an external observer can gain about outcomes of any single
measurement performed by the parties. Finally, we show how the obtained
monogamy relations offer an improvement over the existing protocols for
device-independent quantum key distribution and randomness amplification.
