Can I create my own private templates on Overleaf?

You can create new projects from any of your current projects without publishing them to the Gallery.

[edit] Overleaf V2

Click on the 'copy' action icon on the far right of the project's name on the v2 dashboard.

If you already have the project loaded in a v2 editor, you can click on the Overleaf menu button above the file list, to Copy the Project.

Click on the "Open as Copy" option, from the drop menu next to your project on the Dashboard:

You can further organise your "private templates" using the tagging feature.