BLS says a license is not required for entry-level mechanical engineering jobs. Whether to get one depends on the side of the field you work in.
In product companies and manufacturing, engineers rarely sign drawings that go to a public authority, so few employers ask for a Professional Engineer license. Pay there moves with design responsibility and industry.
In consulting firms that design building systems, someone has to seal the drawings. BLS notes that the PE license allows higher levels of leadership and independence. The path is the same as in other branches: an accredited degree, the Fundamentals of Engineering exam, several years of experience, then the PE exam in the state where you practice.