BRUSSELS—The euro-zone economy is showing signs of sputtering to life, but doubts abound about whether growth can pick up. A big reason economists are skeptical is the huge cargo tethered to the currency bloc's rear bumper: trillions of euros in private-sector debt accumulated over the previous decade that policy makers have done little to address.

In the U.S., households, corporations and banks entered the 2008 global financial crisis with at least as much debt as many nations across the Atlantic. But a mix of measures—including...