Did Franklin County, Virginia issued money in 1862?

A friend has a piece of paper money which is marked on the front:
June 3rd 1862
The County of Franklin Virginia 
Ten Cents
In each corner is the number 10

I want to know if counties were allowed to print their own money in
1862 or before 1862.

Your friend's 10-cent note may very well be genuine. In the 1860s,
fractional currency of less than one dollar was common; many states,
counties, and cities in the Confederacy printed their own notes.

Here's an interesting example: a 90-cent note issued by the City of
Lynchburg, Virginia:

Civil War Preservations
http://www.civilwarpreservations.com/catdet.asp?TargetItem=CR304&CategoryType=Currency

"The beginning of the Civil War saw a number of changes in the money
supply of the United States. These changes eventually led to the
introduction of Postage, then Fractional, Currency. With the advent of
the war and the Government's suspension of specie payments (payment in
gold or silver for the face value of a coin or bill) the value of the
metallic content in circulating coinage quickly became greater than
the face value of the coins themselves. This being the case, it was
actually profitable to melt coins for bullion value. Circulating
coinage rapidly disappeared altogether... Many solved the change
problem by printing their own private notes (scrip, or as they were
known at the time, 'shinplasters.') This practice was most predominant
in the South, where the coinage crisis was the greatest, but was also
widespread in the North. All sorts of companies and organizations
printed private scrip. Scrip was printed by small towns, counties or
even states, by merchants, by railroads, by colleges and even by
individuals."

Currency Gallery: FRACTIONAL CURRENCY BACKGROUND
http://www.currencygallery.org/frac/frac.htm

"Before the American Civil War 1861-1865 the only legal tender was
specie, gold and silver coins from the United States Mint. When the
Southern states seceded these coins, and even copper pennies,
disappeared from circulation into private hordes as a hedge against
the uncertainties of wartime. The Confederate government issued no
coinage and generally declined to print notes smaller than five
dollars, so the state and local authorities filled this vacuum with
fractional currency.

These bills were only redeemable at the County Treasurer's office...
and were worthless outside the immediate area. Currency traders
offered pennies on the dollar to exchange notes on distant places of
uncertain financial status, and often banks would not accept them at
all. The local citizens took them as money because there was nothing
better."

Jefferson County MS GenWeb: American Civil War Money
http://www.rootsweb.com/~msjeffe2/fayettemoney.htm

Google search strategy:

Google Web Search: "fractional currency" + "virginia"
://www.google.com/search?hl=en&ie=UTF-8&oe=UTF-8&q=%22fractional+currency%22+virginia

Google Web Search: "civil war" + "fractional currency"
://www.google.com/search?hl=en&ie=UTF-8&oe=UTF-8&q=%22civil+war%22+%22fractional+currency

Google Web Search: "fractional notes" + "counties"
://www.google.com/search?hl=en&ie=UTF-8&oe=UTF-8&q=%22fractional+notes%22+counties

I hope this information is useful. If anything is unclear, or if a
link doesn't work for you, please request clarification; I'll be glad
to offer further assistance before you rate my answer.

Best regards,
pinkfreud

I am very impressed with the answer and the speed!!