Maine, like New Hampshire, allows permitless carry for eligible adults — so the non-resident permit is mainly about reciprocity value and about being unambiguous if you spend time there.
Issued by the Maine State Police. Massachusetts residents apply as non-residents — you do not need to live in Maine.
The 39 State Multi License Course teaches the law for Massachusetts, Maine, New Hampshire, Connecticut, Pennsylvania and Florida in one sitting as the Massachusetts-only class. You then apply for whichever of the six you actually want.

Six states' course requirements, in one visit, for $349 — instead of $1,249+.
Elsewhere on the site: Getting licensed · Massachusetts gun law · Your town · At the range · Who it's for