I am wondering if I have multiple organizations, each having their own customers (from my understanding, they should be the persons that work in that organization, please correct me if I'm wrong), can the organizations see each other requests? Do they have permission to see just their own requests?
Welcome to Community! To simply put it, different Organizations in the same Project can't view each requests by default. What enables a customer from one organization to view tickets within the same organization is the feature of "share with button" - which allows members of the same organization (customers who belong to or added to the same organization on the Project) to share and view tickets that are created by other members in the same organization.
So for example, a customer creates a ticket and did not share with organization, then only that customer can view that ticket. whereas if the same customer did the same step but shared the ticket with the organization from the list given during creation of the ticket from the Portal, then other customers within the same organization can view that request.
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