Joint estimation and scheduling for sensor networks is considered in a system formed by two sensors, a scheduler, and a remote estimator. Each sensor observes distinct Gaussian random variables, which may be correlated. This system can be analyzed as a team decision problem with two agents: the scheduler and the remote estimator. The scheduler observes the output of both sensors and chooses which of the two is revealed to the remote estimator. The goal is to jointly design scheduling and estimation policies that minimize a mean-squared estimation error criterion. The person-by-person optimality of a policy pair called “max-scheduling/mean-estimation” is established, where the measurement with the largest absolute value is revealed to the estimator, which uses a corresponding conditional mean operator. This result is obtained for independent Gaussian random variables, and correlated Gaussian random variables with symmetric variances. Finally, the joint design of scheduling and linear estimation policies for any two Gaussian random variables with an arbitrary correlation structure is considered. In this case, the optimization problem is recast as a difference-of-convex program, and locally optimal solutions can be found using a simple numerical procedure.