These were struck starting in circa 1858 from the so-called Centered Date obverse die. These were deliberately struck for sale to collectors. Early 2001 articles in Coin World by Dave Bowers discuss practices by the mint in those years.
These were often offered in the pattern sections of sales by Cogan, Woodward and Mason & Company who were well aware of their restrike status. The earliest occurrence for one of these was lot 134 of Cogan's May 1859 Simon Gratz sale.
These were also struck in copper as J132/P159 and nickel J133/P160. Its first occurrence may have been lot 1995 of Cogan's October 1865 F.S. Edwards sale.
Additional information can be found in "Silver Dollars and Trade Dollars of the United States" by Dave Bowers and Mark Borckardt.
Photo courtesy of Bowers and Merena.
Note: At lease 2 examples are known struck over previously struck seated dollars.
An example in ANR's 3/2004 sale was struck over an 1846 dollar. The date area is illustrated below courtesy of American Numismatic Rarities. Although not a good image, you can see a ghost of the 6 behind the 2nd 1.
The other is the famous "1851-O" dollar. It is unclear if this was made using a wrong reverse die or is an overstrike of an 1850-O, 1859-O or 1860-O dollar which resulted in the flattening of the 'O' mintmark. It is likely similar to the other overstrike mentioned above.
We show the date and mintmark area of this piece below.
Date Area of 1851-O
Mintmark Area of 1851-O
For more information on this coin click here.
Photos courtesy of Ira & Larry Goldberg's Coins and Collectibles.