How can I share my form as a template to another user without making it available for everyone else?

You can share the form link to privately. To find the form link, please follow this guide: https://www.jotform.com/help/401-Where-to-Find-My-Form-URL

You can also tell to another JotForm user to clone the form by following this guide: https://www.jotform.com/help/42-How-to-Clone-an-Existing-Form-from-a-URL

If you're referring to the share form template, unfortunately, there's no option to make it private. Here's a guide: https://www.jotform.com/help/157-Sharing-a-Form-in-the-Form-Templates-Gallery