Can those of us living in non-Muslim countries carry out the prescribed punishments of Islam among ourselves if there is an open confession of crime but no official Islamic judge to issue a verdict?
The prescribed punishments in Islamic Law are only to be issued by the judge, since due process and proper procedures of evidence must be observed. They must thereafter only be carried out by properly empowered government officials. Otherwise things will deteriorate into public violence that may bring about dire consequences.
The Muslims who reside in non-Islamic countries in the absence of Islamic courts have to be content to advise one another and exhort each other to righteousness. Wrongdoers may be repremanded by way of pubic avoidance until they show repentance.
May Allah guide us all to what pleases Him.