If $\alpha\in[0,2\pi]$ is the angle from $X_1$ to $X_2$, then the probability that $X_3$ is between $X_1$ and $X_2$, and $X_4$ is between $X_2$ and $X_1$, which is one of the configurations were the chords intersect, will be $\dfrac{\alpha}{2\pi} \cdot \dfrac{2\pi-\alpha}{2\pi}$.
Averaging this for all (equiprobable) values of $\alpha$, we find $\dfrac{\int_0^{2\pi} \dfrac{\alpha}{2\pi} \cdot \dfrac{2\pi-\alpha}{2\pi} d\alpha}{2\pi} = \left.\left(\dfrac{2\pi\dfrac{\alpha^2}2-\dfrac{\alpha^3}3}{(2\pi)^3}\right)\right|_0^{2\pi}=\dfrac16$
The probability that $X_4$ is between $X_1$ and $X_2$, and $X_3$ is between $X_2$ and $X_1$ will also be $\dfrac16$, so the probability that there is an interesection will be $2\times\dfrac16=\dfrac13$, consistent with the intuitive answer given by a symmetry argument.