Yep! All you need is the username and table name and use CartoDB's SQL API like this:
http://{user name}.cartodb.com/api/v2/sql?q=SELECT%20*%20FROM%20{table name}&format=csv&filename={whatever you want}
For instance, I have a table of earthquakes in my account that you can get the CSV for like this:
http://eschbacher.cartodb.com/api/v2/sql?q=SELECT%20*%20FROM%20all_month_3&format=csv&filename=earthquakes
Once you have that URL, you can directly import the file using CartoDB's file importer just as you would for any other file with a URL.
Besides the format CSV, you can do GeoJSON (geojson), Shapefile (shp), and more. See CartoDB's SQL API docs for more: http://docs.cartodb.com/cartodb-platform/sql-api/
Datasets also have public pages like this where you can download in the format you want, etc.