The objective of this paper is to study a class of zero-sum optimal stopping games of diffusions under a so-called Poisson constraint: the players are allowed to stop only at the arrival times of their respective Poissonian signal processes. These processes can have different intensities, which makes the game setting asymmetric. We give a weak and easily verifiable set of sufficient condition under which we derive a semi-explicit solution to the game in terms of the minimal r-excessive functions of the diffusion. We also study limiting properties of the solutions with respect to the signal intensities and illustrate our main findings with explicit examples.