So close yet so far

Denote the angle of the minor arc between the two people as $\theta$ . Note that the distance between the two is given by $10\times\sin{\frac{\theta}{2}}\times2= 20\sin{\frac{\theta}{2}}$ miles. We now integrate this function from $0$ to $\pi$ in order to calculate the expected value. $\int_0^\pi 20\sin{\frac{\theta}{2}} d\theta = 20\times2\times(-\cos(\frac{\pi}{2})+\cos(0))=40$ . Now, to calculate the expected distance, we divide this by $\pi$ , and our distance is $\frac{40}{\pi}$ miles $\approx \boxed{67227}$ feet.