No, you cannot simply drag and drop templates between private and group folders.
Private templates are templates created under the folder My private templates. By definition, private templates are private, and thus can only be viewed by the owner unless the user joins a group that prevents members from having private projects, which will allow the administrator(s) of the group to view these templates after they have been incorporated in Group templates.
Group templates are those created under the folder Group templates. They are visible by default to the group administrator(s). In addition, they can be shared among team members as well.
However, if you want to share contents in a private template to a group, you can:
Go to Notebook.
Click +Add and then select Use template from the dropdown list.
In the lightbox, choose a group project in which you may use this private template, and then click Select project to confirm it.
In the next lightbox, choose the private template which you would like to share, and then click Use template to open it.
Click on the gearwheel to the top right corner of the newly opened entry (i.e. the inserted private template) and select Save as template from the dropdown list.
Give the template a name by typing into the cell and choose Group template folder/subfolder from the list. This shall be the group with which you want to share the private template.
Click Save as new template to confirm. This formerly private template shall be shared with members of the group/subgroup now.