I'm committed to being truly helpful to businesses trying to sort out their online marketing priorities, whether that's how to get started or how to clean up a mess. I'm a provoker and a poker, not so much a cheerleader. I can provide clarity and recommendations on topics including SEO, local search optimization, social media, reputation management, content marketing and more.
For the most part, I concur with the previous answers.
I do want to clarify: There is no Google penalty for duplicate content. A Google penalty is a specific kind of action. Duplicate content, generally speaking, can certainly hurt your sites performance, but there's no penalty.
So much depends on the specifics of your situation. If this other site is the primary source of your revenue (or leads), then you probably want to do what will add gas to that fire.
Again, it depends on your business objectives. Providing them with content, traffic, links, whatever, doesn't provide you any long term "seo equity," but that may be ok.
Couple of additional ideas to consider, if appropriate:
1) Index the pages on your site, but add an rel=canonical tag that points Google to the source of the content (i.e. the huge organic traffic site). You might get a little more traffic. You won't hurt the other site.
2) Come up with a creative way to promote those Q&A's with original content on your site.
So much depends on the specifics of the client's situation. That goes without saying, but shouldn't go without saying.
Most importantly, regardless of the situation, be extremely careful with regards to incentives. This is thin ice with almost any review site that matters. My advice would be not to do it at all.
For most businesses getting 1 review a month would be plenty and, most likely, a huge improvement. With that in mind I would try to create a system/process for the long haul, rather than treating it as a campaign. The only exception to this would be a brand new business or a an established business that had almost no online reviews.
The most challenging thing is getting everyone on the team to own it. This has to be "baked in" to the culture, the systems, the physical environment, etc... (Which remind me, you could incentivize the team to get reviews.) In some situations (re: larger scale) it might be worth using a 3rd party solution that handles the customer follow-up (e.g. GuildQuality.com). For most, I suggest keeping it personal. Have requests come from the owner, store manager or sales person.
In the end, if the service or product is actually bad, then best of luck to you. There's no way to overcome that. Sooner or later the truth gets out.
Feel free to contact me to discuss a specific situation.