Let $A$ be an abelian variety defined over an algebraically closed field, say over $\mathbb{C}$. There is a dual abelian variety $\hat{A}$, along with a Poincare line bundle $L$ on $A\times \hat{A}$. Is there any relation between $\widehat {A\times A} $ and $\hat{A}\times \hat{A}$, for instance are they isogenous. What happens when $A$ is principally polarized, can we say relate the Poincare bundles in this case.

Given two complex tori $X_1$ and $X_2$, there is always a canonical isomorphism $\widehat{X_1 \times X_2} \cong \widehat{X}_1 \times \widehat{X}_2$, see for instance BirkenhakeLange's book Complex Abelian Varieties, Exercise 11 page 43. Indeed let us write $X_i=V_i/\Gamma_i$, where $\Gamma_i$ is a lattice in the complex vector space $V_i$. Then $X_1 \times X_2 \cong V_1 \times V_2/ \Gamma_1 \times \Gamma_2$ and, by standard representation theory of abelian groups, the character group of $\Gamma_1 \times \Gamma_2$ coincides with the direct product of the character groups of $\Gamma_1$ and $\Gamma_2$. 


Over any field $k$, $\hat A=Ext(A,G_m)$ in the abelian category (see "Is the category of commutative group schemes abelian" here on MO) of commutative group schemes over $k$. There is a natural isomorphism $Ext(A \times A,G_m) \cong Ext(A,G_m) \oplus Ext(A,G_m)$ ($Ext$ is a biadditive functor), from which a natural isomorphism $\widehat{A \times A} \rightarrow \hat A \times \hat A$ that you seek . The Poincare bundles on $(A \times A) \times (\hat A \times \hat A)$ and on $\widehat{A \times A} \times (A \times A)$ should be easy to relate as well  just pullbacks via the canonical isomorphisms mentioned above. 


You can define $\hat{A}$ as $\underline{\mathrm{Pic}}^0(A)$. Now, over any algebraically closed field $k$, let $V$ and $W$ be proper (irreducible, reduced) varieties. Then $\underline{\mathrm{Pic}}^0(V)\times \underline{\mathrm{Pic}}^0(W)\to\underline{\mathrm{Pic}}^0(V\times W)$ is an isomorphism. Injectivity is immediate, and surjectivity follows from the theorem of the cube (see e.g. Mumford, Abelian Varieties, section 6 in chapter II). 

