If your horse is sick with a cold, it will cough (greassy cough), usually at the beginning of the work, without any fever or significant despondency.
If the cold remains, the horse could present a little fever (38.5°C for example), the cough will get worse (still a greassy cough) and it could present a small despondency. By the way, the cold is not (a little)contagious.
Usually the cold evolves well on the basis of a good hygiene control (for example hay soaking before distribution, the act of protecting your horse from cold, etc...), a decrease in the work's intensity and quantity as well as the administration of an anti-cough syrup. In the case of a cold which gets even worse, your veterinary will prescribe an antibiotic treatment.
If your horse caught a "general" respiratory illness (as I guess this is your question), like an influenza or a rhinopneumonia, your horse will suffer a strong fever (superior to 39°C), despondency and a dry cough. The cough generally evolves to a greassy cough after a few days which translates an over infection of bacterias. By the way, this viral form is very contagious. There are no specific treatments for influenza types, the best prevention remains a good vaccination. The treatments are symptomatic and an antibiotic treatment limits the bacterias over infections.