In this paper, the authors deal with the linear optimal full-order estimation problem for networked control systems with multiple packet dropouts of both sides from a sensor to an estimator and from a controller to an actuator. The phenomena of packet dropouts are described by two mutually uncorrelated random variables satisfying Bernoulli distributions. A full-order linear optimal state filter and an input filter in the least mean square sense are designed via completing square approach. They employ the received measurements at the present and last moments. The proposed filters are recursively computed in terms of the solutions of a Riccati equation, a Lyapunov equation and a simple difference equation.