"One injured monk and Than Myint Naing, 26, who were being treated for their injuries died from their burns at hospital," it said.
Police imposed a curfew from evening to early morning to control the situation.
"People tried to burn down the mosques," one local police officer told AFP.
The unrest comes amid heightened concerns over Muslim-Buddhist relations in Myanmar, where communal conflict in the western state of Rakhine has left at least 180 people dead and more than 110,000 displaced since June 2012.
An initial report on the police Facebook page on late Wednesday said anger spread after one man was injured during the row in the gold shop.
The report said a mob then descended on the area and destroyed some buildings.
It said six people were hospitalised, and that the Buddhist monk and a Muslim man later died from their injuries. A subsequent police report omitted the religion of the second man.