Competition between groups often involves prizes that have both a public and a private component. The exact nature of the prize not only affects the strategic choice of the sharing rules determining its allocation but also gives rise to an interesting phenomenon not observed when the prize is either purely public or purely private. Indeed, we show that in the two-groups contest, for most degrees of privateness of the prize, the large group uses its sharing rule as a mean to exclude the small group from the competition, a situation called monopolization. Conversely, there is a degree of relative privateness above which the small group, besides being active, even outperforms the large group in terms of winning probabilities, giving rise to the celebrated group size paradox.