We ran a symmetrical base offense, so there were one QB and one FB, two slotbacks, and two receivers. That would be one personnel group. For another group, it would usually be something as basic as a tight end going in for a receiver, so we'd have a wing look to one side. Another group would have two tight ends and no receivers, and so on. Before games, it was the same drill. "Blue group, hands up." Someone would count them. "First subs for blue, hands up". And so on, with red, white, black, and however many offensive groups that we had. Then we'd get the various special teams groups, and so on. Not once did we ever get through that basic setup without there being, "Hold on, how the hell do we have 10 guys for this group?"
During a game, it's not always quite so organized. If a player who's on multiple groups gets injured, his backup might not be aware of all of the roles he's expected to fill. A guy might make a quick sub himself; I remember we played three plays with 10 guys one time because of this situation: the opponent liked to make quick offensive personnel changes, so we were ready with quick defensive changes. We made one of them, and coming off with the guys being subbed out was someone who wasn't supposed to be subbed out. He'd gotten a finger smashed between two helmets and needed a replacement, but he was mixed in well enough that no one picked up on it. And since the offense was in hurry-up mode, the defenders themselves weren't aware of the big gap.
For special teams, there's usually someone on third down yelling at whatever pending special team group to get ready. If there's a long third down, then someone's usually yelling for the punt team to get ready. If there's a chance for a field goal, the FG team is told to get ready. And so on.
During a game situation, whoever is in charge of substitutions is usually carrying on a running dialog with someone in the booth. If he yells out "Black group, get out there", then it's expected that the black group goes out. If someone isn't paying attention, it's on that player. And notice that most of the time, it's a player who's sitting on the bench taking a nice long chug off a water bottle with his helmet sitting about 10' away. The only major sideline rule we ever had was "Be ready to go out there immediately", so we always were off the bench with helmets on.
Frankly, substitution issues are something that's a foreign concept to me, but it's not difficult to see how there can be such issues.