If the widget templates are shared between the sites, any changes that you make to the template will be visible on every site where the widget template is shared.
If you would like to have different templates fro the different sites, please copy the markup of the widget template and then please create a new template under Design -> Widget Templates
in the specified site using the markup you have copied. Please also note to properly select the widget you would like to apply the template to by selecting it from the This template is applied to...
Please refer to the following documentation article
for more details on creating new widget templates.
Do you want to have your say in the Sitefinity development roadmap? Do you want to know when a feature you requested is added or when a bug fixed? Explore the Telerik Sitefinity CMS Ideas&Feedback Portal
and vote to affect the priority of the items