With the Union army...

When camping in position, or defending a position, armies posted "pickets" which were small teams of soldiers spread out and dug into strategic spots. They were sentries of sorts. If the enemy came into their territory, there would be shots fired. They might have to retreat but the rest of the army was alerted that the enemy was near and were they are. It was a dangerous job and they took turns.

On the flip side, when an army wanted to find out where the enemy was, they would send skirmishing teams which were basically small squads. Their main job was to act as a reconnaissance and probe the strength and location of the enemy and try to spot any weaknesses. Again, that was a dangerous job and men took turns. I do know some men volunteered for dangerous work and some tried to get out of it. Sargents were tasked to tried to keep the system fair and the men happy.

I haven't read a lot of direct reports from Confederates on how they operated out in the field but I'd expect it would be pretty much the same thing.