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Organisms have evolved immune systems that can counter pathogenic threats. The adaptive immune system in vertebrates consists of a diverse repertoire of immune receptors that can dynamically reorganize to specifically target the ever-changing pathogenic landscape. Pathogens in return evolve to escape the immune challenge, forming an co-evolutionary arms race. We introduce a formalism to characterize out-of-equilibrium interactions in co-evolutionary processes. We show that the rates of information exchange and entropy production can distinguish the leader from the follower in an evolutionary arms races. Lastly, we introduce co-evolutionary efficiency as a metric to quantify each population's ability to exploit information in response to the other. Our formalism provides insights into the conditions necessary for stable co-evolution and establishes bounds on the limits of information exchange and adaptation in co-evolving systems.