As a former personal injury attorney (Illinois, not Massachusets), I can tell you that if they sue, they will name everyone they can reasonably identify. So if a person is signing the lease, they will be a defendant. All owners on the deed (if it's publicly available) will be named as defendants. They do this because there's no way to know who was really "in charge" of the property and how the injury occurred.
Colleen is correct that there is a difference between who is named as a defendant and what assets are at risk. Being named as a defendant means you have to appear in court. Once there's a judgment, they can try to collect against any assets you own.
While insurance sometimes feels like you're paying for nothing, when you DO need it, you really need it. The insurance company will pay for lawyers to defend you in court (a bigger deal than you think) and will pay any judgment so you don't have to have assets seized to pay a judgment.
