We describe complemented copies of $\ell_2$ both in $C(K_1)\hat{\otimes}_{\pi} C(K_2)$ when at least one of the compact spaces $K_i$ is not scattered and in $L_1(\mu_1)\hat{\otimes}_{\epsilon} L_1(\mu_2)$ when at least one of the measures is not atomic. The corresponding local construction gives uniformly complemented copies of the $\ell_2^n$ in $c_0\hat{\otimes}_{\pi} c_0$. We continue the study of $c_0\hat{\otimes}_{\pi} c_0$ showing that it contains a complemented copy of Stegall's space $c_0(\ell_2^n)$ and proving that $(c_0\ppi c_0)''$ is isomorphic to $\ell_\infty(\ell_\infty^n\hat{\otimes}_{\pi} \ell_\infty^n)$, together with other results. In the last section we use Hardy spaces to find an isomorphic copy of $L_p$ in the space of compact operators from $L_q$ to $L_r$, where $1<p,q,r<\infty$ and $1/r=1/p+1/q$.