When you are sick, moderate exercise is permissible. In fact research shows that people who exercise regularly a get sick less frequently, but there does come a time when you are too sick to workout.
If you do decide to keep working make sure you stay hydrated, over the counter drugs and/or prescription drugs may leave you dehydrated and a little light headed. Be extra careful and make sure you are wiping down machines before and after using and be sure not to touch your face. Keep your eye on your activity level, now isn’t the time to push it.
Urban myth alert — There is no evidence that you can run or sweat out a cold.