What offsets it on the right side? Is the right lung a little bigger?
From the wiki article “Right Lung”
"Difference in size
The right lung, although shorter by 5 cm. than the left, in consequence of the diaphragm rising higher on the right side to accommodate the liver, is broader, owing to the inclination of the heart to the left side. Consequently, the volume, the total capacity and the weight of the right lung is higher than that of the left."
“Inclination” is important. Part of the heart lops over to the right side of the breastbone, though a larger part is on the left. The heart’s also not entirely symmetrical as each of the chambers has different functions. IIRC (~WAG) the left atrium is distended as it provides the initial pump to get the blood through the next few chambers.
Google chest anatomy for a clearer picture.
Why not? I don’t think there’s any real advantage to having the heart on one side or the other. I’m sure if our bodies had developed to have the heart on the right side, that the other organs would have made adjustments just like they did to have it on the left.
Joey P.'s quote suggests a reason: because the liver is so large. The liver is on the right, and the heart is shaped so that it won’t constantly pound on the diaphragm and the liver.
So what about people with their heart on the right side? Is the liver also always misplaced in those cases?
What people have their heart on their right side? 
Right atrium. The left atrium is tiny.
The heart isn’t actually on the left side. It lies on the midline of the body; it’s just asymmetricalwith the largest part on the left. This is partly due to the fact that the left ventricle is much more muscular, since it has to pump blood through most of the body, while the right ventricle only has to pump blood a short distance to the lungs.
People with situs inversuswill have a larger portion of the heart on the right side; the position of other organs will also be reversed relative to the usual configuration.
You’ve never heard of someone having their heart in the right place?