The U.S. dollar plays a key role in international trade invoicing along two complementary dimensions. First, most U.S. exports and imports are invoiced in dollars; second, trade flows that do not involve the United States are often invoiced in dollars, a fact that has received relatively little attention. Using a simple center-periphery model, we show that the second dimension magnifies the exposure of periphery countries to the center's monetary policy, even when direct trade flows between the center and the periphery are limited. When intra-periphery trade volumes are sensitive to the center's monetary policy, the model predicts substantial welfare gains from coordinated monetary policy. Our model also shows that although exchange rate movements are not fully efficient, flexible exchange rates are a central component of optimal monetary policy.