You would need to go into the Users and Roles section and then the section of Service Desk Customer role portion of the service desk project you are working on.
Once there, you can specify the exact customers that are allowed to share out your request in that project by placing their email in that Service Desk Customer role.
After that you would need to go to the "Customer permissions" section in the project settings and select " Any customer, by typing an email address". This setting will only allow users listed in your Service Desk Role of "Service Desl Customers" to send that ticket but it of course would only be able to be sent to another user that is listed in the"Service Desk Customer" Role.
Now if you have random users creating tickets within your service desk, (the customer/ticket can come from any email address), then this isn't possible.
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