This paper considers the problem of communication in the context of strategic information transfer (SIT) concept of Crawford and Sobel in economics. SIT is different from the conventional communication paradigms since it involves different objectives for the encoder and the decoder, which are aware of this mismatch and act accordingly. This leads to a game whose equilibrium solutions are studied here. We model the problem as a Stackelberg game-as opposed to the Nash model used in prior work in economics- where the encoder is the leader and its distortion measure depends on a private information sequence which is non-causally available, only to the encoder; and the decoder is the follower. We consider three problem settings focusing on the quadratic distortion measures and jointly Gaussian source and private information: compression, communication (joint source-channel coding over a scalar Gaussian channel), and the simple equilibrium conditions without any compression or communication. We characterize the fundamental limits -asymptotic in blocklength- of the equilibrium strategies and associated costs for these problems. For the quadratic-Gaussian case, we compute the equilibrium conditions and strategic rate-distortion function explicitly, and show optimality of uncoded communication over an additive white Gaussian channel, paralleling the well-known optimality of uncoded communication in the conventional, nonstrategic communication setting.