This paper studies non-cooperative estimation-privacy games over a network of multiple informed transmitters and one receiver. The transmitters and the receiver have different objectives due to transmitters' privacy concerns which are modeled in the context of the Stackelberg equilibrium of a strategic communication problem. In broad terms, the receiver wants to accurately estimate a random variable, while the transmitters aim to strike the optimal trade-off between providing an accurate measurement and minimizing the amount of leaked information about a private type available to the transmitters. The transmitters, having access to source and type variables, are the leaders and the receiver is the follower. Assuming an entropy based privacy measure on the type variable, a quadratic distortion measure on the source, and jointly Gaussian statistics, we characterize the Stackelberg equilibrium for two different notions of equilibria: I) All transmitters have the identical objective of minimizing the total estimation error subject to an aggregate privacy constraint, ii) Nash equilibria among the transmitters where each one is strategic and aims to minimize its own distortion subject to individual privacy constraints. We show the existence and uniqueness of Nash equilibrium and derive the strategies achieving this unique equilibrium for both notions of equilibrium.