How can the following equation in $x$, $p$, and $s$ be expressed as $x$ in terms of $p$ and $s$? In other words, how can the second reference to $x$ be eliminated from within the nested $\Phi$ and $\Phi^{-1}$ functions?
$$\begin{gather} x = \Phi(\Phi^{-1}(2p - x) - s) \\ 0 < x < 1 \\ 0 < p < 1 \\ s > 0 \end{gather}$$
$\Phi$ is the cumulative distribution function of the standard normal distribution, and $\Phi^{-1}$ is the inverse of the CDF of the standard normal, the probit function.
I agree with the comments suggesting that $x$ is calculated numerically. I also believe that $x$ can be determined graphically. The important question here is whether $x$ exist and is unique. The existence and uniqueness of $x$ can be easily proved by rewriting your equation as $\Phi^{-1}(x) + s = \Phi^{-1}(2p - x)$ and noting that the LHS is an increasing function of $x$ for all $x \in (0, 1)$, while the RHS is a decreasing function of $x$ for all $x \in (0, 1)$ for given values of $p$ and $s$, i.e., there exist a single intersection point between the two curves representing each side depending on the values of $p$ and $s$. Finally, you need to recheck the domains of $p$ and $s$.