There are only two type of users who can make changes in repositories: Administrator and Normal user. Administrators can always do everything they want. Normal users have to have permission to repository to access it.
Normal user can be granted only four type of permission: none, read, modify or admin (really natural). You can set desired permission for certain user and certain repository or when you create a project you can set default permission for all normal users and this permission will be inherited by all repositories within the project.
Returning to your question, if you want to create a repository branch (it's very similar to fork since Kiln creates separated Mercurial repository for each branch), you need to have at least read permission to this repository and modify permission to the project which the repository belongs to.
I believe that the situation with Kiln permissions is similar to the situation on GitHub, since to create your own remote repo fork under your own credentials on GitHub, the repo should be public or you have to have appropriate permission to access it.