Is there a simple way to do this?
At the moment there is no command in MongoDB that would do this. Please note the JIRA ticket with related feature request.
You could do something like:
db.<collection_name>.find().forEach(function(d){ db.getSiblingDB('<new_database>')['<collection_name>'].insert(d); });
Please note that with this, the two databases would need to share the same mongod for this to work.
Besides this, you can do a mongodump of a collection from one database and then mongorestore the collection to the other database.