This study investigates the quantised-input FIR (finite impulse response) system identification under quantised and event-triggered output observations. For the binary-valued quantisation, they introduce a so-called either–or communication scheme to save communication resources. Based on the received data and the triggered indicator, an identification algorithm is proposed to estimate the unknown parameter by use of the empirical measure and weighted least squares. The convergence performance is established and the communication rate is derived. For the multi-threshold quantisation, it can be seen as a vector-valued observation in which each component represents a binary-valued output, and the corresponding identification problem is discussed. A numerical example is included to illustrate the main results obtained.