We consider almost regulated output synchronization for directed heterogeneous networks where agents are nonlinear and non-introspective (i.e. agents have no access to their own states or outputs) in the presence of external disturbances. A purely decentralized time-invariant protocol based on a low-and-high gain method is designed for each agent to achieve almost regulated output synchronization while reducing the impact of disturbances on the output synchronization error. It is also shown that this protocol works even in the case of time-varying graphs.