Singer argues that, just as we're obliged to save a drowning child at modest cost to ourselves (e.g. ruining an expensive suit), so we're obliged to help the distant needy when we're in a position to do so (e.g. by donating to GiveWell-recommended aid organizations). People often balk at this comparison, but I don't see any plausible grounds for escaping the conclusion that we have similarly strong reasons to act in either case.
What's more, I don't think this particular result is really all that counter-intuitive, either. Of course we have incredibly strong reasons to save innocent lives whenever we can! What could be more important, or more worth choosing, than that? This claim about the strengths of various reasons for action -- call it the Act Evaluation -- is eminently plausible.
What is counter-intuitive, I think, is the putative implication that when we fail to donate to effective charities we are thereby just as bad, or as blameworthy, as a person who lets a child drown before their eyes. (Call this the Character Evaluation.) Such a person, we feel, would have to be monstrously callous. As for ourselves, we may not be saints, but at least we are surely not moral monsters. Thus the comparison strikes us as preposterous.
I think this objection to the Character Evaluation is spot on. Consider a quality of will account, on which we are blameworthy to the extent that our actions manifest an insufficient degree of good will (e.g. concern for others). And now notice that differences in what strikes us as salient may lead us to act differently even if there's no difference in our quality of will (or altruistic concern). In particular, our concern for others is much more likely to trigger altruistic action when another's need is made vividly salient to us -- as when we see a child drowning right before our eyes, as opposed to hearing abstract descriptions of the needs of distant strangers.
It seems to be a fact of human psychology that you would need to be a much more callous person to neglect a child drowning before your eyes, than to neglect the needs of distant strangers. I think it is this fact that we are correctly picking up on when we look askance at Singer's analogy. But this fact also shows us why the Act Evaluation does not entail the Character Evaluation, so that our intuitive resistance to the latter should not prevent us from accepting the former. After all, while facts about salience and the psychological vividness might well affect how blameworthy an act of apparent neglect is (since to neglect vivid and salient needs is to manifest a greater callousness than is found in our commonplace neglect of distant strangers' needs), these facts about our own psychologies can't plausibly be taken to affect how choice-worthy the various actions are.
Some confusion may arise due to the ambiguity of 'obligation' talk. Obligation is often understood as closely related to blameworthiness. (Roughly: you're blameworthy if you violate an obligation without an excuse.) Maybe that's the most natural reading. But this moral category ends up being a somewhat convoluted construction that fundamentally concerns character rather than act evaluation. And Singer's argument really only works if we're talking about basic act evaluation (i.e., choice-worthiness, or reasons for action). So perhaps it's unhelpful for consequentialists to speak of 'obligation' here when we're really concerned with the 'ought' of choice-worthiness.
Terminology aside, though, I take it that in practical deliberation we should be concerned with making choice-worthy choices, rather than just avoiding blameworthiness. (Akratic as we are, we might at least adopt the latter standard as a minimum "baseline" that we must meet to maintain our self-respect as moral agents. But it's always better to do better...)
(1) The Character Evaluation is intuitively mistaken, because blameworthiness depends on quality of will, and equally choiceworthy acts might exemplify different degrees of moral (un)concern if the morally relevant features are psychologically much more vivid and salient in one case than the other. In particular, letting a child drown before your eyes plausibly exemplifies (at least in typical human agents) a much greater degree of callousness and lack of concern for others than is involved in our failure to save distant strangers.
(2) The Act Evaluation is plausibly true, since the choice-worthiness of an act depends just on the morally relevant features of the situation, and not on how psychologically vivid and salient these features are to us.
(3) Once we clearly distinguish the Act Evaluation from the Character Evaluation, we may find that only the latter is counter-intuitive, whereas the former is actually quite plausible.
On a more practical note, I hope that by explicitly severing the connection to negative moral emotions (guilt, blame, etc.), the Act Evaluation becomes less apt to provoke defensive responses from people -- You can accept it without thinking yourself a horrible person! Yay! -- and I think it can even start to sound positively appealing. And from there one might be inspired to take some initial steps towards making more of these incredibly choice-worthy decisions, e.g. by joining Giving What We Can or similar philanthropic movements. And that would be cool. Not because you're a moral monster if you don't. But just because it's really worth doing.