Let $\pi$ be the irreducible cuspidal automorphic representation of $\mathrm{GL}_2$. Let $E/F$ be a quadratic extension with given embedding $E^{\times} \to \mathrm{GL}_2(F)$.
For $f_1 \in \pi$, $_2 \in \tilde{\pi}$,
the global version of Waldspurger’s model is defined as
\begin{align*}
    \int_{\mathbb{A}^\times_FE^\times\backslash \mathbb{A}_E^\times}
    \langle\pi(h)f_1, f_2\rangle\, dh,
\end{align*}
where $\langle, \rangle$ denotes the Peterson inner product on $\mathrm{GL}_2(F) \backslash \mathrm{GL}_2(\mathbb{A})$. 

Does this global period integral factor into a product of local integrals over each place 
$v$ of $F$? From the celebrated Waldspurger's formula relating the toric period and the special values of automorphic L-functions, we know this global period integral can be factored into
\begin{align*}
  \frac{L(\pi_E, 1/2)}{L(\pi, \mathrm{Ad}, 1)}  \, \prod_v \int_{F_v^\times\backslash E_v^\times}
    \langle\pi_v(h_v)f_{1,v}, f_{2,v}\rangle\, dh_v.
\end{align*}
The original proof of Waldspurger’s formula relies on the Shimizu lifting and the Siegel-Weil formula.

Is there a direct proof of the factorization of the global period integral into local integrals that avoids using the toric integral of automorphic forms?