It is well known that monetary and fiscal policy are connected by a common budget constraint. In this paper, we study how this manifests itself in the context of the Eurozone, where that connection links the European Central Bank, the 19 national central banks, the Treasuries of 19 countries, and the European Union. Our goal is twofold. First, we wish to clarify how seigniorage flows from the monetary authority to the budget of each country. Second, we seek to answer the question of how the taxpayers of each country are affected by a default of one of the participants to the union. In answering this question, we analyze the mechanisms that ensure (or do not ensure) that net liabilities across countries stay bounded, and we establish how the answer depends on the liquidity premium that each category of assets commands (cash, excess reserves within the Eurosystem, and government bonds). We find that the official risk-sharing provisions of the policy of quantitative easing (QE), whereby national central banks retain 90% of the risk intrinsic in bonds of their own country, only holds under restrictive assumptions; under plausible scenarios, a significantly larger fraction of the risk is mutualized.