We study output-based stabilization of linear time-invariant systems affected by unknown external disturbances. The plant outputs are measured by a collection of distributed sensors, which transmit their feedback information to the controller in an asynchronous fashion over different digital communication channels. Before transmission of measurements is possible quantization is needed, which is carried out by means of dynamic quantizers. To save valuable communication resources, the transmission instants of each sensor are determined by event-triggering mechanisms that only depend on locally available information. We propose a systematic methodology for the joint design of the (distributed) dynamic quantizers and the event-triggering mechanisms ensuring an input-to-state stability property of a size-adjustable set around the origin. Moreover, the proposed approach prevents the occurrence of Zeno behavior on the transmission instants and on the updates of the quantizer variable thereby guaranteeing that a finite number of data is transmitted within each finite time window. The tradeoff between transmissions and quantization is characterized in terms of the design parameters. The method is feasible for any stabilizable and detectable linear plant. The systematic design procedure and the effectiveness of the approach are illustrated on a numerical example.