Opening Word creates a new document based on the normal template. It does not recreate that template in the document. Saving that new document as a template will leave your AutoText in the normal template (assuming that is where it is stored).

If you go to the location where your normal.dot template is stored and copy it (with Word closed) to a new name, you can then put that template in your Word Startup Folder. If you do that, you would want to use the Building Blocks Organizer or Greg Maxey's utility to clear those entries from your normal template (so you don't have duplicates).

If you have not yet saved your AutoText in your normal template, the simple thing to do is to create a new blank template and save it in the Startup folder. Then, when you are in the dialog to save AutoText (building blocks) you can save them directly in that template. Once you have a library of AutoText, you can put it on your network and share it with others. It will need to go into their Word Startup folder.

Normal Template in Microsoft Word - How to Find or Open the Normal Template.

Again, you can use Greg Maxey's utility to backup the AutoText in one template and restore (copy) it into another template.