Motivated by the need to properly address near-term (i.e. interannual to interdecadal) climate prediction as an initial-value problem, intense interest has emerged on the development of data assimilation for coupled atmosphere--ocean global climate models. Basic research on this problem is challenging due to the large computational expense associated with ensemble simulation of coupled climate models over long periods of time. Here we apply an idealized atmosphere-ocean climate model and an ensemble Kalman filter to explore three basic questions on this problem: (1) Under what circumstances is data assimilation needed? (2) Is fully coupled assimilation required? (3) What are the ideal properties of observations for coupled assimilation? Robust solutions for large samples over the model attractor reveal that the slow overturning circulation in the ocean may be constrained with properly handled atmospheric observations alone. Results from the idealized model are compared with those obtained for data from the Community Earth System Model (CESM).