Notes:
The EDU information is based on the best information available.
The printing companies have been listed in shorthand to fit in the space provided. The companies are as follows:
"National" = National Bank Note Company
"Continental" = Continental Bank Note Company
"American" = American Bank Note Company

Although the "secret mark" could be considered a design feature, it is not included in the list of designers in the banner at the top of each stamp. Credit for these marks should be given to Charles Skinner.
The secret marks used to separate the National and Continental stamps can be very problematic. On the lower denominations identification is fairly straightforward, although many collectors still separate the stamps based on color, particularly the two-cent stamp. This is also the method that most use to separate the higher denominations, so pay attention to the colors listed.
"Secret" marks are found on the die proofs of the twenty-four and ninety-cent Continental stamps, which had the rays in the stars strengthened, and the thirty-cent Continental proof stamp, which had a line of shading added to the bottom of the shield. These "secret marks" will be found only on the die proofs.
The so-called secret mark on the fifteen-cent stamp has caused much controversy. It is generally accepted that the apparent strengthening of the "V" in the upper left triangle is found on many, but not all, Continental and American printings. For now, it is probably best to simply sort your fifteen-cent large Bank Notes on hard white paper by color. Any on soft porous paper are of course the American stamp, Scott 189.
Double Paper: The double paper was described by John Luff as being composed of two parts: a "very thin paper, backed by a thicker and firmer one". In theory, the "impressions were to be made on the thin paper which, it was expected, would be destroyed by any attempt to remove a cancellation." (Burns, U.S. Specialist March 1999). This was not always the case, however, since impressions were at least sometimes made on the thicker paper. The collector should bear in mind that the Bank Note double papers are composed of one thick paper and one very thin paper.
The thirty-cent stamp is also interesting. In 1989 Jack Rosenthal published an article in Linn's that showed a clear alteration in the triangular protrusions in the "S" of "CENTS" on some of the American printings, that is Scott 190 and Scott 217. Not all American printings show this alteration, or "secret mark" if you choose, however if the alteration is present the stamp must be an American printing. The American printings were on a soft porous paper making them easily identifiable in any case, but it may help in weeding out Continentals printed on the soft, "transitional" paper. The Micarelli guide lists the Continentals on "soft white" paper, but it is not listed here to prevent confusion.

*Quantity issued:
Estimates are based on available information, primarily from Brookman, "The United States Postage Stamps of the 19th Century- Volume I", and "Linn's U.S. Stamp Facts 19th Century", published by Linn's Stamp News. These numbers are simply estimates based on the best information available and are subject to revision. Please report any updates or corrections via email.

Special Printings:
The 1875 Special Printings were issued to showcase a complete set of the U.S. issues for the first World's Fair officially held in the U.S., at the Centennial International Exposition of 1876 in Philadelphia and also to provide stamps for collectors and others to satisfy numerous requests. Since the stamps of this issue were still in circulation and available at Post Offices, they were not "reissues" per se and are listed simply as "special printings". They are very difficult to distinguish from the regularly issued stamps, other than the very white paper.

Bibliography and suggested additional reading:
Basic Information:
The Postage Stamps of the 19th Century, Volume II, by Lester Brookman (1966)
The Postage Stamps of the 19th Century, Volume III, by Lester Brookman (1967)
The 2011 Scott Specialized Catalogue of United States Stamps & Covers, Scott Publishing Co. 2010
Linn's U.S. Stamp Facts 19th Century, by Elliot A. Landau, editor (1999)
The United States Two Cent Red Brown of 1883-1887, by Edward L. Willard (1970 - 2 vol.)
The U.S. 3c Green 1870-1887, by H. L. Wiley (Severn-Wylie-Jewett booklet #9 - copyright 1915)
U.S. 1887 3 Cent Vermilion, by Dr. H. A. Davis (1922)
The Seven Cent Vermilion U.S. 1871-1873 Issue, by J. W. Sampson(Severn-Wylie-Jewett booklet #42 undated)
Comments on the 10c stamps of 1870-1882, with profile of Jefferson, by Elliott Perry (Collector's Club Philatelist 1949)
Notes on the 10 Cents Jefferson 1870-1879, by Cyril F. dos Passos(22nd American Philatelic Congress)
The United States 15c Stamp of 1870-1890, by William R. Weiss Jr.(1995)

Notable Auction Sales:
U.S. Bank Note Issues, Featuring the Mercedes Collection - Robert A. Siegel Auction Galleries #922 October 19, 2006

Design dimensions:
The Postage Stamps of the United States, John Luff (1902 and 1937)
The design dimensions are from Luff unless otherwise noted.

Colors:
The Encyclopedia of the Colors of United States Postage Stamps - Volume 2 Issues of 1869-1893, by Roy H. White (1981)
The Micarelli Identification Guide to U.S. Stamps: Regular Issues 1847-1934 by Charles N. Micarelli