Local newspaper often have "police blotter" column listing the names of persons arrested by the police and the charges pending. It doesn't violate due process. Those persons still go through the court system. The results of their court trials are later reported in the same papers.
I think the shopkeepers are overstepping their authority. They should detain the accused thief and call law enforcement to arrest them. Provide evidence at the trial and insist on restitution for the theft and any time required to pursue the prosecution. If the accused it acquitted, the false accusation by the shopkeeper should carry a penalty as well. That improves the odds against false accusations. In some cases, the restitution won't be worth the effort to pursue the charges.
If the solution is satisfactory to both the shopkeeper and the thief, why should the effin’ government be involved?