They might be right, they might be wrong. Clearly, most of these critics launch their verbal and written attacks through the lens of partisan politics. The right-wing of U.S. politics has a political incentive to articulate what they see as weak spots in the current president's foreign (and domestic) policies. Whether these criticisms are grounded in reality is irrelevant; the game is to degrade the president's approval. And actually, because of the combination of scant media fact checking, a large presence of low-information voters, and a polarized electorate, there are few mechanisms to deter groups/individuals from putting forward criticisms are more myth than truth.
(Of course, these political maneuvers aren't the sole domain of Republicans. They work both ways in American politics: just as the right engages in exaggeration, worst-case assessments, and tall tales when it's the opposition party, so does the left when it's the party out of power.)
To this point, there has been little evidence that "leading from behind" has fundamentally altered the perceptions of the U.S. What it has done, though, is create opportunities for other countries to advance their interests and goals. Specifically, it has effected an opening for countries like Russia and China to seize the leadership mantle in world politics.
The latest example of this occurred this week. After weeks of America's refusal to step up its involvement in the Syrian conflict, China has recently put forward its own plan to resolve the hostilities. According to Reuters:
Chinese Foreign Ministry spokesman Hong Lei told a regular news briefing that under the "new proposal there are constructive new suggestions such as a ceasefire region by region and phase by phase, and establishing a transitional governing body".
He said it was "an extension of China's effort to push for a political resolution of the Syrian issue".
Sure, China's Syria proposals could be relatively innocuous, maybe even helpful. In a larger, international context, that would be a good thing. But over the long-term, if this situation continually repeats itself, the U.S. could very well suffer considerable harm to its strategic interests and position in the world.
On the one hand, U.S. might find itself squeezed out of its dominant place in the world, having to share leadership responsibilities and duties with countries like China. That, in turn, means that the U.S. will be in a far worse position to defend and advance all the things that are important to Washington and American citizens. On the other hand, as foreign countries assert themselves in international politics, they will seek ends that at times fit with America's worldview and at times run counter to it, the latter of which can jeopardize U.S. interests and values.
So as an example, when China acts as an obstructionist force at the UN or when it cobbles together a proposal that in effect advocates "peace through dialogue"--both of which shield and prop up Bashar al-Assad, the brutal Syrian government and security forces--it's less likely that freedom, democracy, and human rights win the day in Syria. And it's more likely that Syria remains a basket case, a home where conflict, repression, state-sponsored terrorism, and Iranian influence thrives.
Certainly, an economically struggling U.S. has to pick and choose carefully when and where it flexes its power. And leading from behind can carry benefits, such as burden sharing. But as noted above, there are serious downsides as well. A second term Obama administration, or a first term Romney government, needs to think very carefully about employing "leading from behind" as standard practice in American foreign policy.