He—may Allah have mercy on him—was asked:

When invoking blessings upon the Prophet, may Allah bless him and grant him peace, is it better to do so quietly or aloud? Has it been reported that the Prophet, may Allah bless him and grant him peace, said, “Stir your limbs by invoking blessings upon me,” or not? And what of the report attributed to Ibn ʿAbbās that “he instructed them to raise their voices so that those who had not heard could hear”? Please give us a ruling; may you be rewarded.

He answered:

The aforementioned report is a fabricated lie, by scholarly consensus. The same applies to the other report, and likewise to all other reports about raising the voice when invoking blessings upon him, such as those narrated by vendors to promote their wares, or by beggars—storytellers and others—to gather people and collect money from them, and similar purposes.

Invoking blessings upon him is a form of supplication, as the Prophet, may Allah bless him and grant him peace, taught his community when they said:

“We have learned how to greet you with peace, but how should we invoke blessings upon you?”

He replied: “Say: ‘O Allah, bestow blessings upon Muhammad and the family of Muhammad, just as You bestowed blessings upon the family of Abraham. You are indeed Praiseworthy, Glorious. And bless Muhammad and the family of Muhammad, just as You blessed the family of Abraham. You are indeed Praiseworthy, Glorious.’”

This was recorded in the two Sahīh collections. The Sunnah in all supplication is to recite quietly, unless there is a reason for which raising the voice is prescribed.