One of you guys just needs to have a paid membership. It's like $20 but that's for the whole year.
The only other way on this site is to have someone with a membership message both of you in a shared message and then you could message back and forth, but then the person that initiated the group message would see everything you post, so it's not really practical if neither of you has a membership.
Hope that helped "somewhat".
You can also set up a "front" email that you share on your profile and then switch to the primary one you use after one message. Ideally something you delete after the first message relating the primary...that way if any random person messages you (and there are things you want to watch out for online) it's no longer in use, so irrelevant.
If you'd like some info on that a system like that costs a little bit also but there are a few free ones. I know you can do this with Yahoo for free, but I haven't tested the deletion mechanic myself, although it says you can.
I don't see anything wrong with wanting to connect privately like that against site rules, someone correct me if I'm wrong.