Correct. Your local profile can be valid on multiple storage devices (consoles) now. In the past it could only be valid on one storage device at a time. The single valid copy was a reason to keep a profile and all saves on a Memory Unit or USB Memory Unit.
With the restriction lifted and Cloud Storage that's less of a necessity, though as people are finding out Cloud Storage isn't exactly reliable. :-)
IceStorm is correct. In the past (not too long ago) they removed the need to redownload profiles every time you wished to use them on another console. You might have issues with some (not all) content being used on different Xbox consoles because of the
content licenses, but most content such as DLC maps etc won't be affected.