Let $F$ be a local field, let $\omega_1,\omega_2$ be characters of $F^{\times}$, and let $\pi = \omega_1 \boxplus \omega_2$ be the principal series representation obtained via normalised parabolic induction from these two characters. Let $\psi$ be an additive character of $F$ (which I will assume to be unramified) and let $\mathcal{W}(\pi,\psi)$ denote the Whittaker model of $\pi$. When $F$ is nonarchimedean, I'll write $\mathcal{O}$ for the ring of integers of $F$, $\varpi$ for a uniformiser, and $\mathfrak{p} = \varpi \mathcal{O}$.
The $\mathrm{GL}_2 \times \mathrm{GL}_1$ local functional equation states that for any $W \in \mathcal{W}(\pi,\psi)$ and any character $\chi$ of $F^{\times}$,
$$\int_{F^{\times}} W\left(\begin{pmatrix} y & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\right) \omega_{\pi}^{-1} \chi^{-1}(y) |y|^{\frac{1}{2} - s} \, d^{\times}y = \gamma(s,\pi \otimes \chi,\psi) \int_{F^{\times}} W\begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix} \chi(a) |a|^{s - \frac{1}{2}} \, d^{\times}a.$$
Here $\omega_{\pi} = \omega_1 \omega_2$ is the central character of $\pi$.
Via the Mellin inversion formula, it follows that
$$W\left(\begin{pmatrix} y & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\right) = \int_{\widehat{F^{\times}}} \gamma\left(\frac{1}{2},\pi \otimes \chi,\psi\right) \omega_{\pi}\chi(y) \int_{F^{\times}} W\begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix} \chi(a) \, d^{\times}a \, d\chi.$$
See Section 3.2.2 of Michel and Venkatesh's paper "The Subconvexity Problem for $\mathrm{GL}_2$" for this approach.
Alternatively, one can interchange the order of integration and evaluate the integral over $\chi$, yielding
$$W\left(\begin{pmatrix} y & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\right) = \omega_{\pi}(y) \int_{F^{\times}} W\begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix} j_{\pi,\psi}(ay) \, d^{\times}a.$$
Here $j_{\pi,\psi}$ is the Bessel distribution given by the conditionally convergent integral
$$j_{\pi,\psi}(y) = \zeta_F(1)^{-2} \omega_1\omega_2(-1) \omega_2^{-1}(y) |y|^{\frac{1}{2}} \int_{F^{\times}} \omega_1\omega_2^{-1}(a) \psi(ay + a^{-1}) \, d^{\times}a.$$
When $F$ is archimedean, this is a classical $J$- or $K$-Bessel function. This approach is sketched in Section 1.11 of Godement's book "Notes on Jacquet-Langlands' Theory".
Let's return to the former identity. First let me describe the gamma factor. This is
$$\gamma\left(\frac{1}{2},\pi \otimes \chi,\psi\right) = \varepsilon\left(\frac{1}{2},\pi \otimes \chi,\psi\right) \frac{L\left(\frac{1}{2},\widetilde{\pi} \otimes \chi^{-1}\right)}{L\left(\frac{1}{2},\pi \otimes \chi\right)}.$$
Since $\pi = \omega_1 \boxplus \omega_2$, we can write this further as
$$\varepsilon\left(\frac{1}{2},\omega_1\chi,\psi\right) \varepsilon\left(\frac{1}{2},\omega_2\chi,\psi\right) \frac{L\left(\frac{1}{2}, \omega_1^{-1} \chi^{-1}\right) L\left(\frac{1}{2}, \omega_2^{-1} \chi^{-1}\right)}{L\left(\frac{1}{2},\omega_1 \chi\right) L\left(\frac{1}{2},\omega_2 \chi\right)}.$$
When $F$ is nonarchimedean, these $L$-functions are just the usual local Euler factors (i.e. $1/P(q^{-1/2})$, where $q$ is the order of the residue field of $F$ and $P$ is a polynomial of degree at most $1$ with constant term $1$). The epsilon factor is essentially a Gauss sum, since $\omega_i \chi$ is basically a Dirichlet character; see Theorem 2.3.8 of Goldfeld and Hundley's book "Automorphic Representations and $L$-Functions for the General Linear Group" for an explicit evaluation of this.
What about the integral over $\chi$? When $F$ is archimedean, this is almost the usual Mellin transform, since every unitary character of $\mathbb{R}^{\times}$ can be written uniquely in the form $\chi(y) = \operatorname{sgn}(y)^{\kappa} |y|^{it}$ for some $\kappa \in \{0,1\}$ and $t \in \mathbb{R}$: we then have that
$$\int_{\widehat{\mathbb{R}^{\times}}} f(\chi) \, d\chi = \frac{1}{2} \sum_{\kappa \in \{0,1\}} \frac{1}{2\pi} \int_{-\infty}^{\infty} f\left(\operatorname{sgn}^{\kappa} |\cdot|^{it}\right) \, dt.$$
When $F$ is nonarchimedean, we may write each unitary character $\chi$ of $F^{\times}$ uniquely in the form $\chi(y) = \beta(y) |y|^{\frac{2\pi it}{\log q}}$. Here $t \in [0,1]$, while $\beta$ is a character of $F^{\times}$ that is trivially on $\varpi^{\mathbb{Z}}$. The character $\beta$ descends to a character of $\mathcal{O}^{\times}/(1 + \mathfrak{p}^m) \cong (\mathcal{O}/\mathfrak{p}^m \mathcal{O})^{\times}$ for some nonnegative integer $m$, which we denote by $c(\beta)$ and call the conductor exponent of $\beta$. (For $F = \mathbb{Q}_p$, this means that you should think of $\beta$ as being a primitive Dirichlet character of conductor $p^m$.) Writing $\mathfrak{X}$ for the set of all such characters $\beta$, we have that
$$\int_{\widehat{F^{\times}}} f(\chi) \, d\chi = \sum_{m = 0}^{\infty} \sum_{\substack{\beta \in \mathfrak{X} \\ c(\beta) = m}} \int_{0}^{1} f\left(\beta |\cdot|^{\frac{2\pi it}{\log q}}\right) \, dt.$$
