Well, serious attempts at space travel/rocketry didn't really spring up until the early 20th century, when the works of Konstantin Tsiolkovsky and Robert Goddard laid out serious, practical ways in which one could reach outer space.

To be specific,The Exploration of Cosmic Space by Means of Reaction Devices by Tsiolkovsky and A Method of Reaching Extreme Altitudes by Goddard laid out methods in which one could reach Earth orbit using a multistage liquid fuel rocket (Tsiolkovsky) and the possibility of launching a rocket to the Moon (Goddard). Following this theoretical work, a large number of hobby groups arose in the interwar period devoted to trying to develop rocketry. Two of the larger private groups would be the Verein für Raumschiffahrt (Society for Space Travel), also known as the VfR, and Robert Goddard's work. The VfR tried to get funding from members dues(which was difficult during the Depression) and any donations they could get. They even tried to get funding from Fritz Lang to launch a rocket as a publicity stunt for the premiere of one of his films. They also tried to get government contracts to build rockets for the Germany army, and the VfR would eventually go on to form the seeds of the German V2 program. I am not sure if they ever had any serious plans to reach space-the German rocket program did have plans for rockets that could reach Earth orbit, but they only existed on paper-but it does evidence one approach to rocketry outside of a direct government organization-a largely hobbyist group supported by personal funding, donations and possibly even government grants.

With regards to Goddard, he relied on almost exclusively funding from the Guggenheim family, and would head out West to New Mexico to build and launch a series of more and more advanced rockets over the years. His designs never had any serious chance of making Earth Orbit-the highest launch was about 9000 feet-and rarely spoke openly about actually reaching space (largely due to the ridicule he faced after publishing A Method of Reaching Extreme Altitudes),that reaching space was an effort but his approach-a wealthy benefactor funding his work-is another alternative attempt outside of a direct government program.

As for sources, I'm drawing from

This New Ocean by William E. Barrows

The Rocket and the Reich: Peenemünde and the Coming of the Ballistic Missile Era by Michael Neufeld

Rocket Man: Robert H. Goddard and the Birth of the Space Age by David A. Clary

And the excellent documentary series Spaceflight by PBS-narrated by Martin Sheen!(which I spent many rainy days growing up watching it on my pile of VHS tapes).

Hopefully that gives you some clarity!

EDIT: If you want to look at an idea from fiction, in From The Earth to the Moon by Jules Verne, a group of American gun hobbyists decide to see if they can reach the Moon, and plan to build a giant cannon to shoot a bullet into space, carrying passengers. They gain their funds from donations across the world, and use this funding to build their rocket, and launch themselves away. Unrealistic, but it carries the idea that a group of dedicated individuals with technical expertise could basically "crowdfund" their rocket.

If I can use the dozens of pre-manned-space-flight science fiction films I have seen as a source, then I can say that there was a general consensus that space flight would definitely be handled by the government and in many cases seemed to be an offshoot of the United States military. Space pilots are often portrayed as soldiers in a military hierarchy.

Those were movies from the 50's and 60's. If you go much further back in science fiction movies then you're going to see astronauts shot in bullets right into the eye of the moon.