We introduce a new non-zero-sum game of optimal stopping with asymmetric information. Given a stochastic process modelling the value of an asset, one player has full access to the information and observes the process completely, while the other player can access it only periodically at independent Poisson arrival times. The first one to stop receives a reward, different for each player, while the other one gets nothing. We study how each player balances the maximisation of gains against the maximisation of the likelihood of stopping before the opponent. In such a setup, driven by a Levy process with positive jumps, we not only prove the existence, but also explicitly construct a Nash equilibrium with values of the game written in terms of the scale function. Numerical illustrations with put-option payoffs are also provided to study the behaviour of the players strategies as well as the quantification of the value of information.
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