No missing index requests on SQL Server 2012 I am trying to optimize the database by creating missing indexes, but the dm_db_missing_index_details table does not have any data. 

I have the same database on different server and there are many records in that table. So it looks there may be a configuration problem causing the missing index requests to not be saved.

Why are there missing index requests on one server, but there aren't any on the other server (for the same database)?

We have not restarted the database server for the last four days. I am sure there should be index issues, because the same db is running at a different location and that one has many missing index warnings (and we fixed those).
Depending on which version of SQL Server you're on, missing index (and other index metadata) may disappear when you rebuild indexes.

See Kendra Little's post for more information.