While the baseball rulebook features plenty of rules governing the play in and around the dugout, there is no official rule about which team should sit on the first-base side and which on the third-base side. The decision is entirely left to the discretion of the home team.
Despite the lack of an official rule, there is a long-standing tradition of placing the home team dugout on the first-base side. This tradition dates back to the early days of baseball when most players were right-handed. Being on the first base side, allowed coaches to have a clear view of the at bat. This is because the majority of hitters back then were righties.
But while tradition may dictate that the home team sits on the first-base side, it's not uncommon to see them occupying the dugout on the third-base side. Some ballparks, such as Wrigley Field in Chicago, have their home team dugout on the third-base side. This is due to the unique design of the stadium. The third base dugout faces away from the sun during day games whereas the visitors dugout at Wrigley faces direct sunlight during the later innings.
Ultimately, where the home team sits in the dugout is entirely up to them. While there may not be an official rule, the location of the home team dugout can have a significant impact on the game, from strategic advantages to practical considerations such as shade and communication with coaches.