In no way a simple question! First, you ask whether the "opposing sides" in a moral disagreement can "discern the truth" of the moral issue at hand. That raises some terrifically complex questions in moral epistemology, namely, just how do we know moral truths at all? From experience? From the testimony of others? By reasoning? By means of some sort of intuition or perception? Some combination of these? I propose we set those questions aside and focus on some narrower questions about moral disagreement itself: Why do people morally disagree, and is there a suitable way to resolve these disagreements?
Much depends on precisely where the source of the disagreement resides. Let's distinguish four sources of moral disagreement.
Many moral disagreements turn not on moral claims but on disputed questions of fact. For instance, suppose that two people disagree about the morality of capital punishment, one believing it morally justified, the other believing it morally unjustified. However, they may well agree about all the relevant moral claims but disagree about some crucial question of fact (for example, whether capital punishment deters crime). In other words, their moral disagreement doesn't stem from any disagreement about moral values, obligations, etc., but about non-moral matters. They agree about which non-moral questions matter to the morality of capital punishment, but disagree about how those questions are to be answered. Here we can hope that resolving their disagreement about the non-moral question will in turn resolve their disagreement about the moral question.
Sometimes disagreement results when people agree on basic moral claims but disagree about the application or meaning of particular moral concepts. Again, consider the morality of capital punishment. Two individuals might agree that capital punishment is morally justified if it does not treat a criminally cruelly -- but they might in turn disagree about what makes treatment cruel. (You can imagine what this disagreement might revolve around: Is it cruel to deny someone further life? Does it matter how painfully the person dies? Is killing a person any more cruel than having them spend their lives in jail? Etc.) Resolving this sort of disagreement is a bit tougher, but I think we still have reason for optimism. Often times our moral convictions are infused with concepts we use without fully understanding them ('cruel' in this case), and perhaps through some dialogue with others, we can clarify these concepts and thereby resolve our disagreement.
Things get tougher still when the disagreement stems from disagreement about the weight or significance to attach to different values or considerations. Suppose again a disagreement about capital punishment: One believes that capital punishment is wrong because it is cruel, while acknowledging that it is also deserved. The other believes that capital punishment is not wrong because it can be deserved, while acknowledging that it is also cruel. They disagree, then, not about what values or considerations are relevant to the question at hand, but about what weight or significance to assign to them. Here resolving their disagreement seems a bit more difficult, but at the very least, the disputants are working within the same moral frameworks (while disagreeing about what those frameworks entail).
Finally, disagreement is likely to be nearly intractable when the disputants simply do not bring the same set of moral concepts or the same broad moral framework to the disagreement. If the alleged cruelty of capital punishment (or alternatively, its allegedly being deserved) simply does not register with one of the disputants, then it seems hard to envision how they might resolve their disagreement. It's almost as if one or the other of the disputants is simply blind to what, in the eyes of their opponent, is a morally relevant consideration. It's tough to know how the disputants can rationally engage one another in a meaningful way given the source of their disagreement.
So in short, no, moral disagreement is not inherently hopeless. It can often be rationally resolved -- but there is no guarantee of such resolution, and when resolution emerges, it often emerges only very slowly. As with many things, when it comes to moral disagreement, patience is a virtue.