If the wavefunction $\psi(r_1,r_2)$ doesn't represents an entangled state, it is separable: $$\psi(r_1,r_2)=\psi_a(r_1)\psi_b(r_2)$$ In this treatment, we ignore the interaction between two particles so that the initial wavefunction can be written as a product. However, since they are indistinguishable particles, their wavefunction must overlap more or less, otherwise we need infinite potential wells.
My question is: what is the relationship between the interaction of particles and the overlap of their wavefunctions? I know they are not the same thing but I feel confused about their relation. And how both of them affect the indistinguishablility of particles?