It isn't possible to use Timesheets in this way, each user only has one Timesheet per time period.
Unless I'm misunderstanding what you're trying to do, using ETC should give you this forecasting capability. If you enable ETC, it will automatically calculate the remaining work on a task, and can be adjusted by the assignee if they are tracking ahead or behind plan. It is a more precise and mature method of tracking Task completion than % Complete and will likely require some additional training or expectation management for the project team members, but it should do the trick for you.