The exchange bias blocking temperature distribution of naturally oxidized Co-CoO core-shell nanoparticles exhibits two distinct signatures. These are associated with the existence of two magnetic entities which are responsible for the temperature dependence of an exchange bias field. One is from the CoO grains which undergo thermally activated magnetization reversal. The other is from the disordered spins at the Co-CoO interface which exhibits spin-glass-like behavior. We investigated the oxide shell thickness dependence of the exchange bias effect. For particles with a 3 nm thick CoO shell, the predominant contribution to the temperature dependence of exchange bias is the interfacial spin-glass layer. On increasing the shell thickness to 4 nm, the contribution from the spin-glass layer decreases, while upholding the antiferromagnetic grain contribution. For samples with a 4 nm CoO shell, the exchange bias training was minimal. On the other hand, 3 nm samples exhibited both the training effect and a peak in coercivity at an intermediate set temperature Ta. This is explained using a magnetic core-shell model including disordered spins at the interface.