I have created a new JIRA-project using the default permission scheme. When a user, that has the "User" role of the project, tries to create an issue - he gets the error
"User <USERNAME> does not have the 'project' permission"
If adding the user to the adminstrator-role, he can create an issue. So then I read the security scheme and apart from the "Administer project" permission which is only for administrators, members of users has the same rights as administrators.
