The question is in the title. I want to know what happens to $(\chi_G,\chi_H)\in\hat{G}\times \hat{H}$. Are they just passively sitting there as a pair or they give something when applied on $(g,h)$?
My hunch is $(\chi_G,\chi_H)(g,h)=\chi_G(g)\chi_H(h)$ but is this the definition? I may be wrong.
What you have found out is that there is a canonical map $$m : \hat{G}_1 \times \hat{G}_2 \to \widehat{G_1 \times G_2}$$ which maps $(\phi,\psi)$ to the mapping $m(\phi,\psi) : (g,h) \mapsto \phi(g) \psi(h)$. But by itself, $(\phi,\psi)$ cannot act on $(g,h)$, you have to apply this $m$ first. The map $m$ is a bijection, so it's reasonable to view $(\phi,\psi)$ as an element of the dual of $G_1 \times G_2$ anyway.