As there are no pure manganese circulating coins, I'm cheating a bit here. These are manganese alloy coins.

USA, 5 cents, copper-silver-manganese alloy. Diameter 21.2mm, 5g. Found in change.
During World War II, strategic metals such as nickel were in short supply, having been diverted to making parts for the military. So for the four years 1941 through 1945, US 5-cent pieces were made of an alloy of 56% copper, 35% silver, and 9% manganese. The unusual alloy was chosen after a study to determine compatibility with existing copper-nickel coins in pay phones and vending machines.
This is a circulated example, showing the uneven color and laminations typical of these coins due to poor mixing of the metals in the alloy. These coins can be distinguished not only by their color but by the large mint mark over the dome (not visible in this example due to its defect)
USA 1 Dollar. Diameter 26.5mm, 8.1g
According to the US Mint, the new Sacagawea Golden Dollar is
"a three-layer clad construction - pure copper sandwiched between
and metallurgically bonded to outer layers of manganese brass.
This alloy is a golden-colored material composed of:
77% copper
12% zinc
7% manganese
4% nickel. "
As with the Wartime Nickels, the composition of the Sacagawea Dollar was specially chosen to maintain electrical compatibility with vending machines.