If you are interested in this capability using the web-based GUI (not CRM_DNO_MONITOR as suggested by someone else) this scenario is explicitly supported in the Service Provider scenario, where end-users can only see their own tickets and are prohibited from using Queries and other features that might give them access to another user's tickets. More on the Service Provider scenario is in the IMG (transaction SPRO) and also under http://service.sap.com/solutionmanager --> SAP Solution Manager for partners.
If you don't want Queries disabled for end users but still want them to be blocked from seeing another user's transactions, this can probably be achieved using CRM authorizations (a starting point is here: http://help.sap.com/saphelp_crm50/helpdata/en/26/99973915e69238e10000000a11402f/frameset.htm) or something fancy like implementing BAdI CRM_ORDER_AUTH_CHECK. However I am not convinced those will act as filters on Query results (especially those computed by TREX if you are using free-text search) so a user might be able to see the short text of another user's tickets in query results even if they are prohibited from opening the ticket.
Create variant for it. Then create role, assign report for crm dno monitor and then variant. As a result user will be able to view their own message only.
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