Summary: We consider the optimal design of experiments in which estimation and design are performed by different parties. The parties are assumed to share similar goals, as reflected by a common loss function, but they may have different prior beliefs. After presenting a few motivating examples, we examine the problem of optimal sample size selection under a normal likelihood with constant cost per observation. We also consider the problem of optimal allocation for given overall sample sizes. We present results under both squared-error loss and a logarithmic utility, paying attention to the differences between one- and two-prior optimal designs. As asymmetric discrepancy measure features repeatedly in our development, and we question the extent of its role in optimal two-prior designs.