It seems that Organizations could be a very powerful tool in Jira ServiceDesk but it's lacking several fundamental features that could make it really work:

1. Allow on the customer record to show the organization. Right now when we search customer we have no idea if they actually have the organization tied to them. You have to search in two places. 
2. Allow to default an organization for a customer since you can have more than 1 organization per customer. This way when they submit via email or the portal a ticket it will populate an organization by default.
3. Have an admin setting to turn off allowing customers to NOT share with their organization. We have an issue where users see this in the portal and decide not to share it and we have to update them when coming in. Which is difficult because when you search for the customer it doesn't house the organization at their record (see #1).
4. Allow for Organizations to be shared across the Jira platform. We use Software and now have to create a separate field to capture the organization of our Reporter.