The general solution of your equations in a simply connected domain on which $r_2\not=0$ and $r_1\not=\pm1$ is
$$
\beta = \frac12 + \frac1{{(r_1}^2{-}1)}\,
\left(\frac{\partial a}{\partial\theta_1}+b(\theta_1,r_2)\right)
\quad\text{and}\quad
\gamma= \frac12 + \frac1{{r_2}^2}\,
\left(\frac{\partial a}{\partial\theta_2}+c(\theta_2,r_1)\right),
$$
where $a = a(\theta_1,\theta_2)$ is a function of $\theta_1$ and $\theta_2$ only, $b$ is a function of $\theta_1$ and $r_2$ only, and $c$ is a function of $\theta_2$ and $r_1$ only.  

I do not see how you can choose $a$, $b$, and $c$ so that your 'boundary conditions' are satisfied.  Because the general solution depends on three functions of two variables, one would expect to be able to specify (initial) conditions along a surface in the domain, but not along a hypersurface (which is what boundaries generally are).