No, we don't send any information directly to your end-customers as Gelato is a fully white-labelled solution.
At the same time, we forward shipping information along with tracking number to Shopify. If you use Shopify built-in shipping notifications emails, this will also trigger Shopify shipping notification email to your end-customers. If that Shopify shipping notification email template includes tracking number, that will also be included.
For more information, please visit Shopify Help Center.