The U.S. Constitution uses the word dollar in its text but doesn't define it.
The United States dollar (sign: $; code: USD; also abbreviated US$), also referred to as the U.S. dollar or American dollar, is the official currency of the United States of America and its overseas territories. It is divided into 100 smaller units called cents.
The U.S. dollar is the currency most used in international transactions and is one of the world's dominant reserve currencies. Several countries use it as their official currency, and in many others it is the de facto currency. It is also used as the sole currency in two British Overseas Territories, the British Virgin Islands and the Turks and Caicos islands.
Are people free to create their own currencies?
Soveriegn states have the authority to create their own currencies. However, "some" currencies created by a country does not have value outside of their country.
Does the U.S. Dollar have a special standing?
Yes, it's the world's dominant currency, and used as the currency for oil trades. However, that is changing as China becomes a greater producer of energy.
Beyond that, many countries not considered US territories use the US dollar as their sole currency.