Hi Michael,
Sunil is right that duplicate content can hurt your SEO. However, I'm not sure how exactly how it would effect site A. You would probably have a harder time getting site B to rank since site A is already established for the duplicate content and has more backlinks from other sources.
In any event, having unique content for each site would be the best bet to establish & keep good SEO for both sites. Perhaps you could start with your homepage & main category pages and work through the most critical areas first since the site is so large.
I found a great article about duplicate content posted today at Search Engine Guide that you might find helpful:
Theories in Duplicate Content Penalties - Search Engine Guide Blog