This is again false. The geometric interpretation is as follows: given étale morphisms $Y \to X_1$ and $Y \to X_2$ of affine schemes, the image factorisation
$$\Gamma(X_1,\mathcal O_{X_1}) \underset k\otimes \Gamma(X_2,\mathcal O_{X_2}) \twoheadrightarrow \Gamma(X,\mathcal O_X) \hookrightarrow \Gamma(Y,\mathcal O_Y)$$
corresponds geometrically to taking the (scheme-theoretic) image $Y \to X \hookrightarrow X_1 \times X_2$ of the product map $Y \to X_1 \times X_2$. If $Y$ is smooth, then so are the $X_i$, but $X$ need not be smooth. If $Y \to X$ is surjective, it cannot be étale \[Tag [07NG](https://stacks.math.columbia.edu/tag/07NG)\].

We can turn this around to make a counterexample: start with a map $Y \to X_1 \times X_2$ of smooth schemes such that the image is singular but both projections $Y \to X_i$ are étale.

**Example.** This is the typical picture of a plane nodal curve: let $Y = \mathbf A^1 \setminus \bigl\{0,\pm\tfrac{\sqrt 3}{3}\bigr\}$ and $X_1 = X_2 = \mathbf A^1$, and consider the morphism $Y \to X_1 \times X_2$ given by $t \mapsto (t^2-1,t^3-t)$. The image lands in the nodal curve $\{y^2 = x^2(x+1)\}$, only missing the points $(-1,0)$ and $\bigl(-\tfrac{2}{3},\pm \tfrac{2\sqrt 3}{9}\bigr)$ where the tangent line is horizontal or vertical (hence the choice of $Y$). Thus the compositions $Y \to X_1 \times X_2 \to X_i$ are étale, but the image
$$X = V\bigl(y^2-x^2(x+1)\bigr) \setminus \bigl\{(-1,0),\bigl(-\tfrac{2}{3},\pm \tfrac{2\sqrt 3}{9}\bigr)\bigr\}$$
of $Y \to X_1 \times X_2$ is singular at $(0,0)$.

(This parametrisation is explained on [this page](https://www.math.purdue.edu/~arapura/graph/nodal.html) of Donu Arapura, which is literally the first result that my favourite search engine produces when I type 'nodal curve'.)

Translated back into algebra, this means $S = k\bigl[t,\tfrac{1}{t},\tfrac{1}{3t^2-1}\bigr]$, with $R_1 = k[t^2-1] \subseteq S$ and $R_2 = k[t^3-t] \subseteq S$.