No perversion involved. MGL C. 41 S. 98 states in part:Carrying on the badge is a perversion of a vague statute that deals with issuing equipment, it does not get you an LTC. It's basically a wink and a nod kind of thing.
ETA citation: https://malegislature.gov/Laws/GeneralLaws/PartI/TitleVII/Chapter41/Section98Section 98. The chief and other police officers of all cities and towns shall have all the powers and duties of constables except serving and executing civil process. They shall suppress and prevent all disturbances and disorder. They may carry within the commonwealth such weapons as the chief of police or the board or officer having control of the police in a city or town shall determine; provided, that any law enforcement officer of another state or territory of the United States may, while on official business within the commonwealth, carry such weapons as are authorized by his appointing authority.
On duty or off duty for MA LEOs, if their chief authorizes his/her officers to carry a certain type of weapon they are GTG.
They can't buy anything on the badge legally in the state however, but they can possess anything within the category authorized by the chief (not just the gun that is issued).
Understand that this also means that a chief can revoke the authority to carry/possess a weapon on the badge if he/she decides to (like in the case of someone under suspension, under investigation, unable to work, etc.).