That's a more complicated question  than it sounds.

First there is a difference between a Word template and a sharepoint  template in
a document  library. Word Templates are .DOT or .DOTX files and only work when
located on the user's local machine. They are not used by SharePoint, but you
could create a document library that would contain Word Templates for download
and use. SharePoint Word templates are .DOC or .DOCX files that are associated
with a document library through the use of a content type. This is what
SharePoint uses when you click on the New button in a Document Library that has
been created to house Word documents. The SharePoint template file is stored in
a Forms folder inside thedocument library. When you click the new button
SharePoint starts Word on your workstation and downloads the associated
SharePoint template to Word. But if you do a SaveAs on the document it will
default to the document library itself, not the Forms folder. It will also not
use the name of the template, but will default to the more generic Document1,
document2, etc just like Word when you start a document using a locally
installed Word Template.

Even if the user  tried to change the location  to the Forms directory and
overwrite the SharePoint template that would be limited by security  settings.
So having access  to a portal, even write  access does not mean that you can
overwrite the SharePoint template associated with a document library.

Didn't know that there was another way to do that. I¹m
wondering if the document  Library would then be the best way. Sounds like
it would if you have Word documents, but we also have InDesign template
files that need to be accessible for our users. It doesn¹t sound like you
could do a Sharepoint template document library with InDesign files. Am I
correct?

This only works for Office documents. You can however set
Read Only permissions  on the InDesign template files so that people can't
overwrite it.

You can create a new document  library that uses Word as its template. Save your
Word 'template' as template.doc on your desktop. OPen the new document library
in Explorer View and then open the Forms folder. Replace the template.doc with
your template.doc. Now go back to standard view of your library and click on the
'New Document' button and Word should open using your template.

That doesn't stop users from saving documents to the library that don't use your
template or overwriting  any documents that are in the library; but, your
template should not get overwritten unless they open the library in Explorer
View, open the Forms folder, and replace the template.doc.