Intrinsic defects give rise to scattering processes governing the transport properties of mesoscopic systems. We investigate analytically and numerically the local density of states in Bernal stacking bilayer graphene with a point defect. With Bernal stacking structure, there are two types of lattice sites. One corresponds to connected sites, where carbon atoms from each layer stack on top of each other, and the other corresponds to disconnected sites. From our theoretical study, a picture emerges in which the pronounced zero-energy peak in the local density of states does not attribute to zero-energy impurity states associated to two different types of defects but to a collective phenomenon of the low-energy resonant states induced by the defect. To corroborate this description, we numerically show that at small system size N, where N is the number of unit cells, the zero-energy peak near the defect scales as 1/lnN for the quasi-localized zero-energy state and as 1/N for the delocalized zero-energy state. As the system size approaches to the thermodynamic limit, the former zero-energy peak becomes a power-law singularity 1/|E| in low energies, while the latter is broadened into a Lorentzian shape. A striking point is that both types of zero-energy peaks decay as 1/r2 away from the defect, manifesting the quasi-localized character. Based on our results, we propose a general formula for the local density of states in low-energy and in real space. Our study sheds light on this fundamental problem of defects.