We examine and compare the local stress autocorrelation in the inherent states of a fragile and a strong glass: the Kob–Andersen (KA) binary mixture and the Beest–Kramer–Santen model of silica. For both systems, local (domain-averaged) stress fluctuations asymptotically reach the normal inverse-volume decay in the large domain limit; accordingly, the real-space stress autocorrelation presents long-range power law tails. However, in the case of silica, local stress fluctuations display a high degree of hyperuniformity, i.e., their asymptotic (normal) decay is disproportionately smaller than their bond level amplitude. This property causes the asymptotic power law tails of the real-space stress autocorrelation to be swamped, up to very large distances (several nanometers), by an intermediate oscillatory-exponential decay regime. Similar contributions exist in the KA stress autocorrelation, but they never can be considered as dominating the power law decay and fully disappear when stress is coarse-grained beyond one interatomic distance. Our observations document that the relevance of power-law stress correlation may constitute a key discriminating feature between strong and fragile glasses. Meanwhile, they highlight that the notion of local stress in atomistic systems involves by necessity a choice of observation (coarse-graining) scale, the relevant value of which depends, in principle, on both the model and the phenomenon studied.