jmc

If $a$ is a root of both polynomials, then $a$ is also a root of the difference of the polynomials, which is $$(x^3+7x^2+px+r)-(x^3+5x^2+px+q)=2x^2+(r-q)=0.$$And if $a$ is a root of this polynomial, so is $-a,$ and their sum is 0.

By Vieta's formulas, the sum of the roots of $x^3+5x^2+px+q=0$ is $-5,$ so the third root is $-5.$ Similarly, the third root of $x^3+7x^2+px+r=0$ is $-7,$ so $(x_1,x_2)=\boxed{(-5,-7)}.$