Maybe. In a recent divorce case in New York, the court permitted a woman to serve her husband with divorce papers via Facebook. Is this dependable and fair to the person being sued? Can anyone do this? It depends. In this particular case, the court only allowed it when the husband is nowhere to be found, all other methods have failed, and he is notified by phone call and text repeatedly that this is happening. This is not much different than "service by publication" which typically has been done by putting something in the Public Notices section of a newspaper, and only can be used if the court is satisfied that all other reasonable methods have been tried and failed.
The fairness part is all about the guarantee of “due process of law” in the U.S. Constitution and most state constitutions. Whether it is for divorce or any other reason, due process guarantees that the person being sued be notified of the suit, what the suit is about, and be given a chance to defendant him or herself in court. The best method of “serving” the person with a lawsuit is in person, usually by a law enforcement official or licensed process server. In certain types of cases, or if the person cannot be served personally, service by mail is allowed. But if all personal service and service by mail, a court may allow service by “publication” in a newspaper or any other method considered reasonably reliable.
In the New York case cited above, the husband could not be located, had left his apartment without a forwarding address, has no DMV record, and has no fixed place of employment. However, he kept in touch with his wife from time to time with his wife on Facebook. So, under these circumstances, the court permitted the wife to serve her husband with divorce papers via personal message on Facebook, provided that she also took certain additional steps to make sure he was aware of the lawsuit for divorce that was sent to him. Actually, under the circumstances of this case, the method used to notify the man of the lawsuit probably is more dependable than the more traditional notice by publication in a section of a newspaper that no one ever reads. Therefore, the method used to serve the husband in this case probably is sufficient to comply with the constitutional guarantee of due process of law.