Check out this forum discussion to see some expert discussion on the topic of centered, non centered, and partially centered parameterizations.
As I understand it, centered actually works better when you have informative data (large N relative to \sigma) for a particular group, while non centered is better for uninformative data (small N relative to \sigma) for a particular group. The thread I linked also discusses trying to center some groups and not others depending on how much data is in each group, but it seems like finding the cut off for N is a difficult task.
Also check out this paper for more in depth analysis of the challenges hierarchical models present for Hamiltonian Monte Carlo.