We study the problem of loss estimation that involves the choice of a first-stage estimator of the parameter of interest, incurred loss, and the choice of a second-stage estimator of this loss. We consider both a sequential version where the first-stage estimator and loss are fixed and optimization is performed at the second-stage level, and a simultaneous version with a loss designed for the evaluation of the estimators of the pair composed of the parameter and the loss together. We explore various Bayesian solutions and provide minimax estimators in both cases. The analysis is carried out for several probability models (multivariate normal, Gamma, Poisson, negative binomial), and relates to different choices of the first and second-stage losses. The minimax findings make use of a least favourable sequence of priors and depend critically on particular Bayesian solution properties, namely cases where the second-stage estimator is constant.