By default, only user groups that have the 'Manage thread by anyone' permission (found under Forum Moderator Permissions) can change thread titles.
However, there is a paid add-on that will allow you to assign 'Edit own thread title' to any user group you desire.
You may also request such a feature to be made into a free add-on here.
There is no permission for allowing posters to change their thread title in core xF, it is a moderator permission, not a user permission and global for the node / entire forum, not just for own threads. I.e. there is no way to do this without an addon.