I'd like to address this comment from the question:
I do not think the discussion about our research about our common
ancestors is of interest to anyone else, nor is it suited for the Q/A
design of this site.
Whether your research is of interest to anyone else is a different issue from wanting to conduct your collaboration in private. Your research ...
I can empathise, as you will see in my answer to How strict should we be with Not An Answer flags/closure on new users?
I would encourage both you and your cousin to include contact details in your user profiles like I have done in mine. You can tailor your profile to each site that you participate in.
I think it is OK to use comments to direct each other ...
Stack Exchange sites are designed around a principle that (with few exceptions) all communication between its users should be within the site and use its focused Q&A format supplemented by comments to help clarify those posts.
There is a Meta SE Q&A on How do I contact other users? that may help you find other ways to do it, but in general we would ...