Expanded from an earlier answer
Although it is difficult to extrapolate wider American public opinion, there was some awareness in press that the massive flocks of passenger pigeons had disappeared by the turn of the century. This does not mean however that people connected the disappearance to a man-made extinction, between the 1870s and 1900, commentators floated several possibilities for the birds' disappearance.
One popular theory is that the birds responded to the hunting by moving to locations unknown. An 1877 report from the magazine Auk would claim:
Every bird lover would rejoice to hear that this wonderful bird had finally outwitted its great persecutor and lengthened its lease on life by "going West" in the true American spirit of liberty; and it is fitting that we use every effort to protect and foster a movement so unprecedented.
In an American Poultry Journal report on a sole passenger pigeon's 1895 roosting in Chicago's Lincoln Park, "passenger pigeons have gone from the West, excepting one or two places in the Indian Territory, as completely as they have disappeared from the East," but the journal's editorial staff noted that the correspondent was in error and immense flocks were still in the West and Michigan's lower peninsula. There were scattered reports of the bird making their way to Chile or even as far as Australia. Yet these reports were unsubstantiated and could not be confirmed. One theory was that the birds had perished in a vast transpacific or transamerican migration away from this new predator. Although these commentators acknowledged the existence of mass hunting, they tended to shift the blame onto the birds' own migratory habits for the final disappearance.
Once it became apparent that the passenger pigeons were not returning or had found a new roost, some commentators began to use the word extinction, but often with reluctance. The American Poultry Journal report began:
Once upon a time, and and it is hard for a bird lover to begin a story this way for it presages a tale of extinction.
In the nineteenth-century, human-caused extinction was actually a relatively novel concept. Although humans had pushed many species to extinction, scientific study of animal populations was a relatively recent phenomena. Even when it was clear that human predation was at fault, there was a strong tendency for some commentators to blame the animals for their own extinction. Scientific entries on the dodo often emphasized the role of its own stupidity in its demise. The naturalist Charles Dixon's book Lost and Vanishing Birds condescendingly observed that the dodo was "unsuspicious, tame, defenceless, and even stupid." Although Dixon lamented the demise of the dodo, he couched it in terms that blamed it on an thoughtless human society. With regards to the passenger pigeon, Dixon noted that "the rapid decrease of the passenger pigeon must be attributed to the direct persecution of civilized man," but that determined efforts by naturalists can preserve the species. In a letter to the journal Recreation inquiring about the pigeons' whereabouts, the editor replied with resignation "that it is sad that so noble a bird such as this should be exterminated, but such seems to be his fate." This sort of reply fused elegiac incredulity and condemnation of man's profligate husbandry of the earth's wildlife, but did not fundamentally question the fundamental righteousness of humanity's domination over the natural world.
The death of the last passenger pigeon Martha in 1914 led to the passenger pigeon becoming a symbol of for the growing conservationist movement. One popular rumor that circulated within the press during 1915 was that Department of Agriculture had offered a $10000 reward for a nest and eggs, which its 15 June newsletter denied.
But the rumor persisted and conservationists would invoke the passenger pigeon as an example for some kind of regulatory system for wild game. Larry St. John, the author of the Chicago Tribune's "Woods and Waters" sporting and nature columns asserted:
No one in those days [the 1860s] believed that the passenger pigeon would ever be decreased, much less exterminated- they had no example like we have. As we will point out in this column from time to time, other birds are going the way of the passenger pigeon- slower, but just as certain. Think it over.
The time has passed when marketing game in this country can be looked upon as a business. The only game that should be sold should be raised by carefully supervised and licensed game farmers. Anything more liberal than this means the end of game in the United States and soon. Game dealers and market hunters have killed their own goose; now let them pay the price.
William Temple Hornaday, one of the early generation of American conservationists, explicitly cited the passenger pigeon’s fate in an appeal to save the Sage Grouse:
The passenger pigeon millions went the same way, by commercial slaughter, so quickly that no one realized what was happening until it was all over! [emphasis original] For ten years, the American people refused to believe that all those millions of birds have gone forever; but now they know that this is true.
Forest and Stream’s 12 September 1914 coverage of Martha’s death concluded that her tale was a cautionary one:
If only the human species could emulate this commercial spirit, act in unison for bird-protection without commercial quibbling, curb its mania for bird adornment, check excessive “sport for sport’s sake,” and annihilate potting for market, some of our threatened birds would reestablish their slender hold and escape their impending extinction. In the early settlements, pigeons, turkeys, parakeets, and Heath hens were plentiful; civilization and culture came; the hills and valleys were deforested, the lowlands were cultivated; in short, the balance of nature was excessively disturbed, yet where we collectively provided these original occupants refuge, or how have we restrained ourselves, to promote their greater increase, when they were most rapidly lessening? The conscience balm has always been “They will be ever common.”
Finally, the Florida Audubon Society reported in its July 1919 Bulletin that the Migratory Bird Treaty was explicitly to prevent the type of activities that caused the demise of the passenger pigeon:
The bunch of “sports” who last February near Tallahassee killed 1399 Mourning doves in one day on a baited field, and quit the murder only when they ran out of shells, have learned that the Federal Government will not permit this species to be wiped out, as the Passenger pigeon was wiped out, to make a vandal’s holiday. The hundred dollars they each paid for the information that the Migratory Bird Treaty Act is a living proposition, always to be reckoned with, was much too little.
As a whole, Martha's tale gave a degree of focus to the problem of extinction caused by human hunting and many conservationist literature used her fate to promote regulations.
Sources
Anonymous. "Passenger Pigeon Takes Refuge in Lincoln Park," American Poultry Journal. Vol. 26: No. 1(1895), 15-16.
Anonymous. "Wild Pigeons in Kentucky." Recreation. Vol. 10, No. 2 (Feb. 1899), 138.
Dixon, Charles. Lost and Vanishing Birds; Being a Record of Some Remarkable Extinct Species and a Plea for Some Threatened Forms. London: J. Macqueen, 1898.
Florida Audubon Society. "Paying for Their Sport." Florida Audubon Bulletin, Vol. 1, No. 11 (July 1919): 2.
Hornaday, William T. Save the Sage Grouse from Extinction; A Demand from Civilization to the Western States. New York: New York Zoological Park, 1916.
John, Larry St., "Woods and Waters: Something to Think About." Chicago Daily Tribune, January 2, 1917: 14.
_. "Woods and Waters: Something to Think About." Chicago Daily Tribune, January 23, 1917: 16.
Wright, Albert Hazen. “Martha the Last Passenger Pigeon Dead: The Passenger Pigeon: Early Historical Records, 1534-1860.” Forest and Stream. Volume LXXXIII, No. 11 (September 12, 1914) :336-338.