Does this analysis also hold in a monetary union like the Eurozone? The answer is yes, provided we structure the bond-buying programme carefully.
Suppose the ECB buys an amount of government bonds of Greece. This leads to interest payments from the Greek government to the ECB, which then distributes the profit it makes to the member countries according to the equity share of each country in the ECB (Germany 27%, France 20%, etc.).
Such a bond purchase therefore implies fiscal transfers from Greece to the other member countries. However, there has been an agreement within the ECB to return the Greek interest payments to the Greek government. Thus we reach the situation obtained in a standalone country discussed in the previous paragraphs. That is, the Greek bonds held by the ECB lead to a circular flow from Greece to the ECB and back to Greece. The ECB could easily stop this circular flow by writing off the bonds. This would have no effect on the other member countries.
Note that this is also the rule under the QE programme. The government bonds (80%) bought by the ECB are put on the national banks’ balance sheets. As a result, a circular movement of interest payments is organised from the government to the ECB and back to the same government.