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)} $.