We give a construction of two-sided invariant metrics on free products (possibly with amalgamation) of groups with two-sided invariant metrics and, under certain conditions, on HNN extensions of such groups. Our approach is similar to the Graev's construction of metrics on free groups over pointed metric spaces. - See more at: http://www.ams.org/journals/tran/2014-366-12/S0002-9947-2014-06010-8/#sthash.ullL9RMw.dpuf