The short answer is that it was in the best interests of the army for soldiers and commanders alike to know who was in charge of any given group at any given time. Even if the army hadn’t furnished the names of people who had been promoted to the press, soldiers would have been informed of their commanding officer (C.O.), and that information would have inevitably got out to the press (so why even try to hide it?). Yes, it might be in one’s best interest to keep the enemy in the dark about what’s going on in your army, but it would have been catastrophic for an army to operate without the full knowledge of who was in command at any given time. In other words, the benefits outweighed the risks.

Even so, it would be practically impossible to keep this info secret. In the United States military, during the Civil War as now, precise records are kept to denote when an officer is promoted to general for purposes of rank distinction. For example, if two brigadier generals are in a command situation, technically, the one promoted at an earlier date outranks the other. Thus, promotions and new assignments were publicized at some level in most instances to keep people properly informed. Besides, many of the officers in the Union and Confederate armies knew each other via West Point or because of other pre-war schooling or business opportunities. When one of them got promoted, if the officers themselves didn’t hear about it through mutual friends, they might get word of it via a letter from home, or just by reading the newspapers, where a familiar name might jump out at them. For the most part, though, for colonels and above, messages from the war department or from superiors informed fellow officers of familiar (or otherwise) appointments. For higher appointments, in the Civil War (and I believe today as well), when the president appointed a Major General, that appointment had to be approved by Congress, so it was not something that could be kept secret.

And from a practical level, besides the familiarity these generals had with each other (so their natural interest in knowing how acquaintances were doing, i.e. scuttlebutt), it was in a general’s best interest to know who was in command of a divisions or corps in the area. How Stonewall Jackson handled his troops differed considerably from how Richard Ewell handled his, and whether you were a Confederate or Union general, this information might be vital to how an attack might go off. Likewise, if we’re going to stick with Gettysburg as an example, how Gen. Warren handled his defensive arrangements versus how Gen. Sickles handled his shows how knowing which commander is in charge can tell you a lot about how they’ll react in a critical situation. Civil War generals were extremely interested in who was in command, and where, and even if this information wasn’t readily available via gossip and the newspapers (which it was), it would have been to their advantage to find out (which, again, wouldn’t have been that hard). Soldiers were kept informed of command changes, and it wasn’t too terribly difficult to get information out of privates who often had nothing but time on their hands to chat, write letters, and otherwise socialize.

[Sources: Bruce Catton, ‘Grant Takes Command’ & ‘Grant Moves South’; James McPherson, ‘Battle Cry of Freedom’]