This paper considers nonlinear systems with full state feedback, a central controller and distributed sensors not co-located with the central controller. We present a methodology for designing decentralized asynchronous event-triggers, which utilize only locally available information, for determining the time instants of transmission from the sensors to the central controller. The proposed design guarantees a positive lower bound for the inter-transmission times of each sensor, while ensuring asymptotic stability of the origin of the system with an arbitrary, but priorly fixed, compact region of attraction. In the special case of Linear Time Invariant (LTI) systems, global asymptotic stability is guaranteed and scale invariance of inter-transmission times is preserved. A modified design method is also proposed for nonlinear systems, with the addition of event-triggered communication from the controller to the sensors, that promises to significantly increase the average sensor inter-transmission times compared to the case where the controller does not transmit data to the sensors. The proposed designs are illustrated through simulations of a linear and a nonlinear example.