Absolutely! That is the idea of the Workgroup Templates folder. See Templates in Microsoft Word for information on the folder structure for the templates.

The folder will need to be designated as the Workgroup Templates folder on each user's computer. This can be done through policies (but I don't know how to do it). Again, this can be a shared network folder but my preference is to copy the contents to a local workgroup templates folder as a part of the user login process.

In Word 2010 File > New > My Templates is not all that bad; I prefer just going directly to my templates with Ctrl+N or a click on a QAT icon. Note that Word 2013 throws new stuff at you under File > New. I have been using macros and keyboard shorccuts to get me straight to the My Templates classic dialog since Word 2002 rather than messing with all of the online templates that I don't use and the intermediate steps MS has interposed to show them off. See File New Variations in the Versions of Word to see how this has changed over the years and ways to make it work better for you.