New company – same or different domain name for intranet/internet

New company. Naturally want to do things right.
Company name is LONG …i.e. domain name is long.

Trying to figure out an intranet naming strategy.
xxx.int.xxxxxxxxxxxxxxxxxxxxx.com?
Or do we register a different domain name – and use that —> xxx.xxxxx.com
(marketing wants us to use a convention – so that they have the flexibility to use anything in the xxx.yyyy.yyy (x) position.

1. are there downsides for using the same domain name on intranet and internet
2. if “same” is the way – what are the trends in naming conventions?

thanks
jill