Dispute resolution procedures vare greatly from dispute to dispute, and one payment method to another.
When a dispute is raised, the escrowed cryptocurrency being traded can only be released by a dispute resolution manager. It is up to this manager to determine who the crypto is released to.
The first thing that happens when a dispute is raised, is a resolution manager will look through the terms and conditions of the trade, and chat between users. This way the manager can get an overview on whether payment details have been shared, payment has been claimed to have been made, and any other information shared. This is why it is very important to discuss all aspects of the trade and payment details in the chat on the LocalCoinSwap platform, and not on any 3rd party communication app.
From there the manager may request evidence from each user regarding the payment. The evidence required will depend on the payment method used. This, and possibly further evidence will be used in order for the dispute resolution manager to come to a determination on the outcome.