After you have made the changes in the first part of mscdex's answer you might need to add the host to the list of known hosts - before the npm install
command will work.
You can do this by cloning the private repo to another directory:
git clone ssh://git@bitbucket.org:private/private-module.git
You might be asked if you want to proceed, type yes
and enter, then bitbucket.org is trusted. Go back to your project directory and retry npm install
. This is what was needed for mscdex's answer to work for me.
There are other ways of adding trusted hosts, but this does that in addition to verify that you can actually get to the desired private repo.