I am not a fan of duplicating content on other sites. It can be done, but why give away your own work? That is how I feel about things.
Be that as it may, if you chose to, you must get assurances that the other site will place a canonical tag in their copy of the page pointing to your original content. Do not share your content without this.
Also, make sure they attribute your work with an author profile and actual link to your content/site. Make sure this link does not have nofollow in it. Do not share your content without this.
Get your agreement in writing. Do not skip this step. Type up a basic agreement, a letter of understanding, with all the criteria you expect them to provide.
Otherwise, without these things, there is no up-side for you.
On the positive side, having content with a canonical tag to your content, attribution, and a link to your site can boost your sites importance in the search engines. However, like you said, the nature of things may change.
What you may lose in search to this site could positively effect your site in other ways. For example, people following the links to your site to see what else they can find.
Also do not share all of your content. Just some of it. Save something for yourself. How much you decide. Perhaps keep half or two thirds depending upon how much you have. If you do this, your site may see an increase in search results to the content you chose to keep. Who knows for sure?
Lastly, think about what specifically you are willing to share. Do you have unique content that exists no where else? Perhaps you will want to keep this content and share simpler more common content that you do not compete well for anyway.
Before making the decision, research this other site thoroughly. Get as much information about ranking, do not use Alexa, but do check other performance sites and see where the other site stands. Check to see that you can call not only the individual you are talking to, but any other contact information in whois and on the site itself and see if you feel you can trust them. Do your homework before saying Yes.