It seems clear that if you're going to have a single monetary zone with a single central bank, that you need some kind of common banking framework. The absence of such a framework is one of several major institutional weaknesses of the eurozone. But as Europe talks about forming a "banking union" there's considerable ambiguity as to what that would look like. For example, this manifesto of German-speaking economists in favor of such a union first tries to reassure its audience that the banking union doesn't mean German taxpayers are going to be stuck with the bill for Portugese banks (emphasis added):
In no way does this endorse a collectivisation of bank liabilities. Rather it is essential to cede key powers of regulatory intervention in member countries to a banking supervision authority at the European level. This European banking supervision authority should have the ability to authorise rapid recapitalisation of troubled banks. In extreme cases, this may mean the expropriation of previous equity holders and the partial conversion of bank debt into equity. A unified resolution procedure must be capable of recapitalising, unwinding, or liquidating insolvent financial institutions in an impartial manner.
But then it seems to me that it does in several ways endorse a collectivization of bank liabilities (emphasis, again, added):
At the same time, creditors must be made liable for risky investments, so that the resolution of troubled financial institutions can be executed without taxpayer money. In order to secure the financial stability of a banking union, a common restructuring fund that can intervene and impose binding conditionality on reorganisation plans is needed. The ESM can play this role. A stronger Europe-wide deposit insurance system can also contribute to the long run stability of the banking system.
That's collective responsibility for bank liabilities if you ask me. And this is one of the several ways that the American fiscal and monetary union works. Texas had a localized property bubble in the 1980s and it when it unwound tons of Texas banks went bust. But the deposits were insured by the FDIC which created a large net flow of money into Texas and helped cushion the blow.
Now it's easy enough to understand why German (and Finnish and Dutch and so forth) taxpayers are not thrilled about this idea. But it's how a unified deposit insurance system works and it's one of the main ways through which you can say it contributes to stability.