GitHub: Is it possible to collaborate with someone on a private gist?

No. Your friend should fork the repo, and make her changes there.

You can then merge in your friend’s commits by:

1. Clone your gist repo locally
2. Add your friend’s fork as a remote
   1. e.g. if your friend is named Cindy: git remote add-url cindy https://gist.github.com/cindy/df03bdacaef75a80f310
3. Fetch your friend’s commits: git fetch cindy/master
4. Merge your friend’s changes into your repo: git merge cindy/master
5. Push the changes back to GitHub: git push origin/master