In Beat! Beat! Drums!, from the very first time one reads it, there's obviously something complicated going on with the speaker. On the one hand, he's calling (perhaps cynically) for the drums and the bugles to continue beating and continuing to interrupt the daily activities and lives of the people in the country. The sounds are symbols of war, both the actual physical violence and the intensity and tension that it brings to the civilians. On the other hand, the speaker is criticizing the drums and bugles, calling them "shrill," "terrible," and even ordering them to drown out the cries of children to their mothers.
The question is about what Whitman's speaker represents, and what the drums and bugles represent. This particular collection of poems was published in 1865, so if Beat! Beat! Drums! wasn't written during the war it was at least written immediately afterwards. Neely says that Whitman viewed the war as a battle for unity, which could mean he favored the North and was opposed to the South. In this view, it would make the most sense for the drums and the bugles to represent the Southern armies and their push for separation. They go out and interrupt the lives and northerners who are trying to go on with their daily lives, and divide them into one side or another and force them to become involved when they don't want to. This interpretation would suggest that Whitman thinks only the South wants to fight, and the North just wants to continue as it is.
However, Neely's point could also mean that he simply wanted the war to end and be over with, and was opposed to the violence in general and not a particular side. This view is strongly supported by Beat! Beat! Drums!. One basic detail is that Whitman is talking about multiple drums and multiple bugles; he could just as easily have replaced every instance of "drums" or "bugles" with a "drum" or "bugle" and the rest of the poem would have been almost unaffected. Instead, he consciously chose to use plural. This suggests that, rather than one side being the aggressor (the South), drums are coming in from all sides to bring the country into conflict. Multiple elements are coming together to bring about the war, and neither of them is justified, at least in the speaker's mind.
Also noticeable is that Whitman only talks about the sounds of the instruments. Again, the speaker could have ordered the drummers to keep marching or to hold along the line of battle, which would have associated them a specific side or the other, but he did not. By leaving them as only sounds, they're more ambiguous, as if the North and the South aren't really that different from each other. Their sounds don't have any specific direction, but just spread their influence and their message to everyone. The beating of the drums interrupts all different kinds of life, including the rich, the poor, peaceful people, entertainers, businessmen, farmers, and they even attempt to drown out the sounds of the traffic.
There's no specific motive behind them, nor a specific purpose in front of them. They aren't fighting for slavery, against slavery, or because of economic struggles. Whitman sees them only as a voice whose message doesn't matter, because no matter how justified, the result is only going to be violence and war. His cynicism to the drums is saying that whether you're right and they're wrong, or you're wrong and they're right, it's neither of you that matters. It's the ordinary people, the ones that don't want to fight, that will be most affected. They'll be forced to choose a side, and their choice will divide them and their country in half. Neely is right, as Whitman doesn't want the war to be won by the "right" side, he just wants the war to be over so that all of these people from all their different walks of life can go back to being unified.