Thanks for the followup.
And to answer your question, yes and no.
If the plugin is on the old site and you set up the redirect to go to the new site, then you can set up redirects for the one to the other (but not using wildcards, only a specific link, i.e., http://www.oldsite.com/my-page/ to http://www.newsite.com/my-page/)
If you have the plugin set up on the new site and the the link comes in because the domain was forwarded from one to the other (or resolved from DNS A Record), then yes (although at that point, you would not need the plugin as the server would receive the correct URL parameter and just redirect to the right place, assuming they are exactly the same on both sites).
If you trying to use it to have the anything from one folder on the old site to go to the same folder on another site, then no. You would need to use a domain forward, a DNS 'A' Record or simply use .htaccess.
The plugin is really designed to help make old 404 links go to a new page (and not give the page not found 404 error), as well as move existing pages to another link if needed - like if you had a page called http://mysite.com/monthly-special/ you could redirect it each month to the correct page with the special with only needing to give out the one original link to everyone.
The original reason for making the plugin was to make a top level page for a page menu drop down and make it go to a category or sub page. Since then WordPress has developed their Navigation Menus which allow you to do it without the plugin any longer - but since people still use the plugin in so many different ways, it is still very handy and popular.
I know I was rambling, so I hope I answered your question appropriately. If not, let me know.