Only about 30% of victims have fevers, and those may last as short as one day. True asymptomatic cases have no symptoms at all. But they still shed as much virus as people who later develop symptoms. This is because the viral infection causes no symptoms at all. All the symptoms come after the viral replication phase, when the victim's innate immune system starts attacking the virus. One of the first responses is for Natural Killer T-cells to see foreign protein on the surface of cells, and release some poisons to kill those cells. These poisons (besides killing the infected cell) cause fever and attract more white cells to the area.

Of course, one NK T-cell killing one infected cell is not going to cause a fever. It takes a lot of that to make one. It's a race: If you have a small viral inoculum, your immune system can take care of it and you never know. But you still spread the virus, because that's what evolution has created in the virus. Spreading is it's sole "purpose in life". The virus sees no benefit in killing you, or even making you sick other than sneezing or coughing.