Parallel cluster-randomized trials can identify mean effects by site characteristics and risk differences between fixed predictors, while leaving residual heterogeneity and absolute prediction risk (mean squared error of site-effect predictions) unidentified. We acquire missing same-site cross-arm information by randomizing participants within selected sites. Under randomized site selection with compatible participant reference distributions and intervention versions, we characterize necessary and sufficient observation patterns for quadratic targets in a model with unrestricted joint distributions of site response means. From finite independent participant samples, we construct unbiased moments and exact covariances. Under regularity conditions, we prove consistency as informative paired site counts grow, allowing fixed within-site sample sizes. For a fixed predictor, we optimize paired-site selection and unequal cell counts using corrected risk variances, pilot error, and costs. In an International Stroke Trial information-deletion experiment, pilot allocation reduces the conditional variance of reconstructing the complete-information corrected risk estimate by 34.3\% relative to uniform allocation with equal expected cost. In 40 Honduras villages, correction reduces estimated residual variation from 0.04081 to 0.00282 in squared log-consumption units for the same fixed explanatory basis; the explained share rises despite a smaller explained component, illustrating how participant sampling noise can inflate estimated residual variation.

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