Fevers are not an indicator of whether or not an illness, including a cold, is contagious. Illnesses are contagious during their incubation period and that depends on the illness. Whether an illness if even contagious depends on what the illness even is.
With a cold, people are contagious 24-48 hours before symptoms even show up. That's your incubation period. Then once symptoms show up you're contagious for 3 days but can have symptoms for up to five. However, there are hundreds of strains of the virus, so you could still be contagious the entire time until you're no longer excreting bodily fluids. Fever plays no part in contagion.
So here's the thing. If your child has a cold and is coughing, there are mucus and saliva droplets spewing all over. Ditto with sneezing and nose blowing. If your child has drippy boogers, they'll get smeared onto surfaces with hands even if they're using tissues and washing hands.
The courteous thing to do when you or your kids have a cold is to keep them away from parties and play dates throughout the duration of the cold. If you have to send your child to school because technically there's no fever and they haven't had one for 24 hours then make an effort to suppress their cough and minimize the boogers. Teach them to cough into their elbow, use tissues to blow their nose, and wash their hands frequently.
If my child is miserable, I will keep her home even without a fever. A fever isn't the only indicator that they should stay home.