I hate to burst your bubble but you were not watching Sputnik at all.
What you were seeing was the rocket booster that launched the Sputnik into orbit.
Sputnik could not be seen with the naked eye.
Sputnik could, too, be seen by the naked eye. Satellites shine with either the moon or sun on them. You’d only see the rocket flare while it was putting the satellite into orbit. Then it separates from the satellite and, as it loses its orbit, eventually disintegrates in the atmosphere.