According to what I know, resources are not allowed to change the Start and Finish columns. If you need to keep the data input by resource, my suggestion is to use the "Actual Start" and "Actual Finish".
To do so, there are two steps:
1. In Project Professional :
1.1 start Project Professional and logon to server
1.2 Open the project you wnat to adjust
1.3 in the menu, click Tool \ Customize \ Published Fields. select "Actual Start" and "Actual Finish" from the "Available fields" area into "Fields in the Tasks View" area. make sure that the "Let resource change field" checkbox are checked for both the two columns.
1.4 Republish Assignments
2. In Project Server
2.1 login to Project Server as Administrator
2.2 go to Admin \ Manage Views and modify view for Timesheet
2.3 select "Actual Start" and "Actual Finish" from the "Available fields" area into "Displayed fields" area, save.
Hope this helps you.
Ingrid