There is no official rule regarding home team dugout location. The home team dugout can be found on either side throughout MLB. I can tell you that for as long as I can remember, the Indians always were on the 1st base side at home. Later for some reason, they switched to the 3rd base side. Peter Gammons once explained that teams east of the Mississippi had their dugout on the east side (first base side) of the stadium. While teams west of the Mississippi would have their dugout on the west (third base side) Today some teams with new ballparks have changed this early tradition. Peter Gammons was not correct. The Chicago Cubs and the Chicago White Sox -- both teams in the Major League since it's inception both have their home dugouts on the third base side. The Colorado Rockies and the San Diego Padres have always had their dugouts on the first base side.
The Atlanta Braves dugout is on the 1st base side of the field.
The Royals' home dugout is on the first base side.
The Rogers Centre.
The dugout on the first base side is always home no matter what league you are in.
The Mets' dugout at Citi Field is on the first base side.
3rd base side
Along the Third Base Line
The first-base side. Or, depending on the league, the home team gets choice... but the home team picks the 3rd-base side almost 99% of the time.
The first base side of the field.
Yes several teams do. The Red Sox dugout is and always has been on the first base side to name one.
Interestingly, there is no official rule about where the home dugout has to be. And the White Sox are not the only team to have theirs on the third base side-- so do the Tigers and the Indians (and a few others too). Some historians think having it on the third base side goes back to baseball's early years, when many players were also third-base coaches; after standing on the third base line, it was thus much easier to just run back to the dugout and sit for a few minutes. But that explanation may be just another baseball legend.