$\DeclareMathOperator\Cent{C}\newcommand\oG{\overline G}\newcommand\oe{\overline e}$Put $\oG = \operatorname{PGL}_n(\mathbb C)$. I hope you will permit me to denote the involutions by $\oe_i$ instead of $e_i$.
You have indicated that you are interested in the case where $\oe_1$ and $\oe_2$ belong to a common torus $\overline T$, which we may as well assume is the diagonal torus in $\overline G$. This is a partial answer addressing this case, where it no longer matters that our elements are involutions.
In this setting, if we lift $\oe_i$ to elements $e_i$ of the diagonal torus $T$ in $G \mathrel{:=} \operatorname{GL}_n(\mathbb C)$, then $\Cent_G(e_i) = \Cent_G(e_i)^\circ \to \Cent_{\oG}(\oe_i)^\circ$ is a surjection, and $\Cent_G(e_i)$ is the full pre-image in $G$ of $\Cent_{\oG}(\oe_i)$. Therefore, $\Cent_{\oG}(\oe_1)^\circ \cap \Cent_{\oG}(\oe_2)^\circ$ is the image of $\Cent_G(e_1) \cap \Cent_G(e_2) = \Cent_G(e_1, e_2)$. So the question becomes whether $\Cent_G(e_1, e_2)$ is connected. Most of the rest of this answer deals with quite arbitrary $z$-extensions of algebraic groups, but here I use a fact particular to $\operatorname{GL}_n$.
We begin with the (general, not specific to $\operatorname{GL}_n$) fact that the map from $\operatorname{stab}_W(e_1, e_2)/\langle s_\alpha : \alpha(e_1) = \alpha(e_2) = 1\rangle$ to $\Cent_G(e_1, e_2)/\Cent_G(e_1, e_2)^\circ$ is an isomorphism, where $W$ is the Weyl group of $T$ in $G$. Now, specialising to $\operatorname{GL}_n$ and so identifying $W$ with $\operatorname S_n$, an explicit computation shows that $\operatorname{stab}_W(e_1, e_2)$ is the product of permutation groups that respect the decomposition of $\{1, \dotsc, n\}$ into maximal subsets $I$ such that all diagonal entries of $e_1$ with entries in $I$ are equal, and all diagonal entries of $e_2$ with entries in $I$ are equal; and that this is just $\langle s_\alpha : \alpha(e_1) = \alpha(e_2) = 1\rangle$. Thus, $\Cent_G(e_1, e_2)$ is connected, so $\Cent_{\oG}(\oe_1)^\circ \cap \Cent_{\oG}(\oe_2)^\circ$ is the image $\Cent_{\oG}(\oe_1, \oe_2)^\circ$ of $\Cent_G(e_1, e_2) = \Cent_G(e_1, e_2)^\circ$, as required.