You do not need to PUT the template to the site. When you make a change to a template, DW should prompt you to propogate the changes to the child pages. After you accept that prompt, you need to then PUT the child pages to the server.
When you save the Template, it should automatically prompt you to update all local files. Then you have to upload all changed files. If for some reason the local files didn't get updated when you saved the template, go to Modify - Templates - update pages