Assuming the FULLNAME() is the same as the employee name in the filter you could create a calculated field [EmployeeName] = FULLNAME(). Put that in the filters shelf and set it to TRUE. Then the user will only see their report on opening.
You can try something like this. In server tab there is a option to filter based on users. Lets assume I've users based on Product Categories. You can restrict other users to view other categories once you assign.
See the screenshot.
Select your respective users who will access only Furniture then clikc OK and Publish the workbook.
Here I've given privileges to All Users who can only view the data for furniture category.
I hope this will give you some idea.
The above suggestions are better if you want to limit a user's view to themselves.
If you don't want to limit the user's access, then you can use a self/others parameter to swap between a worksheet that shows only the user's data, and a second worksheet for other user's data. Save the dashboard with the parameter set to 'self' and each user will open the dashboard to their own data. Tableau Public