Can a person serve another person a summons through the mail or does it have to come from the court/sheriff's office?
In most cases no, but there are exceptions (for instance in cases by publication a copy is mailed). Since you gave us no details as to the type of case or how it affects you, there is no way to answer you.
Sheriff; private process server; publication; secretary of state; FedEx or UPS; etc etc
Glen is correct, we need more information