If your coworker only has read access to the repo, then he'll need to fork the repo, clone it down, make his changes, push them back up, and then make a pull request from his repo to yours.
Obviously you'll have your reasons for restricting permissions, but if he has read-write access to the repo, he simply needs to make a branch on that repo, push it, and he can then issue the pull request.
Here is some additional info from bitbucket on working with pull requests