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We can’t punish someone unless we are certain and have evidence of the theft. Even if we have the evidences, we don’t have the authority to punish, the punishment has to be by judge or Hakim-al-Shar’.

For a theft to be proved, two just witnesses should have seen the thefty . It may also be that there is just one witness but the owner also testifies that robbery has taken place. The thief may himself confess twice of his theft and deserve to be penalised. If he confesses only once, the stolen goods are taken away from him and restored to the owner. He is not punished for theft.
If the owner takes back his goods or allows the thief to keep them before the matter is reported to the Qazi and does not press for a penalty, the thief is not punished. However if the crime is proved before the judge even the owner cannot save the thief from punishment.
Some Mujtahids believe that if two just people have not witnessed the theft and it is only proved by two confessions of the thief, the judge has the prerogative to condone him. It is mentioned in the book Tahzib that a person came to Amir ul-Mu’minin Imam Ali (as) and confessed of having stolen something. Imam Ali (as) asked him: “Can you recite some portions of the Qur’an?”
He said, “Yes, Surah al-Baqarah.”
‘Imam Ali (as) said: “I have respited your hand in exchange of Surah al-Baqarah.”
Ashath said, “O ‘Ali (as)! Have you overlooked the Divine penalty?”
‘Imam Ali (as) replied:  “What do you know? Awarding the penalty is only necessary when two just witnesses have testified. But if the crime is confessed by the thief himself, the Imam can condone him.”
If the crime satisfies the above conditions, the amputing of the hands could only be carried out by the just ruler or judge. No other person is qualified to award the penalty or carry out the punishment. The Qazi is responsible for recovering the stolen goods and restoring them to the owner. If the goods are used up or lost, the thief has to make up for the goods lost. The order for the returning of stolen goods is the same, whether the theft is proved according to Shari’a or not. If a thing belonging to someone else is taken, it has to be returned.
https://www.al-islam.org/greater-sins-volume-2-sayyid-abdul-husayn-dastghaib-shirazi/twenty-third-greater-sin-stealing

May Allah(swt) grant you success

Wassalam,

Syed Haider