Mongo Management Studio provides 3 ways to copy a collection.
Copy a collection within a database
A collection is copied within the same database. Only the name of the destination collection has be entered.
Copy a collection to another database on the same MongoDB server instance
The collection is copied within the same MongoDB server instance to a different database. The name of the collection is kept and therefore must not exist in the destination database.
Copy collection into another MongoDB server instance
Copies the collection into any database on the specified destination server instance of MongoDB. If this database does not exist, it is created automatically. Here again, the collection is not allowed to exist in the destination database already.
If an authentication via userSource is needed for the copy operation on the MongoDB server instance, the field "login database" offers the ability to specify the appropriate database to use for authentication. The authentication data can also be specified within the connection string for the destination server.
When using this functionality you need to take care of MongoDB's naming conventions and make sure a database with the same name does not already exist on the server.