No, you do not need a local database. The error you are getting could be a number of problems.
You should make sure that the connection is correct to the database, and make sure that the server is properly configured so your system can reach it with correct permissions.
However, if you reached the database to add the tables to your app, and the error reporting a local database is the wrong connection, my guess may be that the publish portion of the project is where the connection string would be generated.
If this is the case I would look at changing the connection string for testing, which is discussed here: how to change the connectionstrings of a lightswitch application in debug mode