Osgood was born in Rochester, New Hampshire, the oldest child of a family of watchmakers. The family moved to California in 1888 and he went to study in Santa Clara and San Jose. He joined in the activities of the Cooper Ornithological Club and found company in Chester Barlow and Rollo H. Beck. He taught at a school in Arizona for a year and then moved to the newly formed Stanford University where he came to meet Charles H. Gilbert and David Starr Jordan. He joined the staff of the Bureau of Economic Ornithology and Mammalogy, of the United States Department of Agriculture at the age of 22. This group later became the Bureau of Biological Survey under Clinton Hart Merriam.