Hi
I have had sites myself with shared content and the end result was that neither of them ranked at all. They were set up in a pre-Penguin world (before 2011) and when the update really cut in September 2012 I lost 60% of my traffic in one day.
I have also worked on many sites who shared content across their own pages resulting in the same collapse in SERPS - You can read about the biggest mistake that website owners make here: https://moz.com/learn/seo/duplicate-content
In certain circumstances, you can share others' content by way of syndication. You'll see it on MOZ occasionally. They will have produced a great article and at a later date will share it across some other article sites as the authority will have been established as a MOZ article. Note that these are small'ish articles, not whole sites.
What you are talking about is basically, willingly, creating a duplicate site to site A. If you do that your rankings on site A will fall and site B will never gain any rank at all if the content pages are duplicates.
Yes, a competitor could damage your site if they were so inclined. Negative SEO is the practice of sharing your content to a number of sources thereby creating mass duplication. While Google should recognise yours as the original that is rarely the the case.
Duplicate content is at the very core of SEO. If someone is telling you differently then they are wrong.
However, it is your website and I would completely agree with your strategy of playing devil's advocate. If it was my site I would want as much corroboration as possible. So go and ask other SEOs but make damn sure they know what they are talking about and it isn't a 'bloke down the pub! because it can cost you hugely.
We probably lost £½m through our own naivete - never again!
Regards Nigel
