Both options have their pros and cons. 
Anonymously has as advantages an example I that it might be less frightening to enter because no one can see it. A disadvantage is that you can't see if anyone is entering. You might think you are the only one working on a story and that might work discouraging.

If the contest mod were to actively keep track and post when an entry is submitted on the prompt&entry thread everyone would know that people are entering, how many stories there and that it is an active contest. Then I think anonymously would loose a major con, making it the better of the two options. 

Not knowing who you write your feedback for or who you vote for does seems to me to be a more honest and unbiased way of feedback. 

Also, the discussion thread is important, even if entries are submitted trough pm, to keep interest. Not everyone asking questions or discussing possibilities will enter and some will read, not write in the discussion thread and still enter.