First, this just sounds risky and likely to lead to accidental removal or merging of code. I think a better scenario would be for the first person to create the branch and push it up, then the second person to pull the branch down and begin working from there.
However, I suppose ONE WAY to do this is that the second person can fetch down the history and then merge the remote branch into their local branch and then push their branch up.
git fetch
git checkout branch1
git merge origin/branch1
git branch -u origin/branch1
git push
From this point forward the second person should have their local branch tracking the same remote branch and fully merged and integrated.