In Gelman and Hill's Data Analysis Using Regression and Multilevel/Hierarchical Models, they present a very compelling idea of 'random' effects offering a kind of compromise between no-pooling (i.e. including an indicator for every group variable and estimating a distinct intercept for each) and complete pooling (not including any group level predictors and thus forcing groups to have the same intercept). They say that including a 'random' intercept term (i.e. allowing the intercept term to have its own probability distribution) is an ideal compromise - as it serves to shrink group level effects towards the population level average when the group level data is scarce or noisy.
A common use of the 'random intercept' model, however, is in longitudinal studies in which the same individual is measured multiple times. In this case every group is a single individual. Here I don't see how the variance in the group level intercept can possibly be estimated - and the idea of 'partial pooling' no longer makes clear sense to me. Can anyone clarify the distinctions in the way random intercepts are understood between these two contexts? If a random intercept model in a longitudinal study with subjects measured at multiple time points can't be understood as 'partial pooling', how should they be better understood?