This paper proposes event detector and controller co-design criteria for sampled-data systems in which static or dynamic output feedback controllers are used. The resulting criteria provide sufficient conditions to ensure the asymptotic stability of the closed-loop system with reduced communication rates among sensors, controllers, and actuators. The transmissions are mediated by event detectors at both sensor and controller nodes. For static output feedback control, the sufficient condition is given in terms of the feasibility of bilinear matrix inequalities (BMIs). The BMI feasibility problem is converted into a nonlinear optimisation problem involving linear matrix inequalities (LMIs), which is solved via the complementarity linearisation algorithm. For dynamic output feedback control, the sufficient condition is formulated into an LMI feasibility problem, which is readily solved with existing tools. Numerical examples are included to show the effectiveness of the proposed methods.