We study the growth of two competing infection types on graphs generated by the configuration model with a given degree sequence. Starting from two vertices chosen uniformly at random, the infection types spread via the edges in the graph in that an uninfected vertex becomes type 1 (2) infected at rate $λ_1$ ($λ_2$) times the number of nearest neighbors of type 1 (2). Assuming (essentially) that the degree of a randomly chosen vertex has finite second moment, we show that if $λ_1=λ_2$, then the fraction of vertices that are ultimately infected by type 1 converges to a continuous random variable $V\in(0,1)$, as the number of vertices tends to infinity. Both infection types hence occupy a positive (random) fraction of the vertices. If $λ_1\neq λ_2$, on the other hand, then the type with the larger intensity occupies all but a vanishing fraction of the vertices. Our results apply also to a uniformly chosen simple graph with the given degree sequence.