<p>Has anyone come up with a good way to share a set of Visual Studio Code Snippets (actual .snippet files that Visual studio uses) amongst a team of developers automatically?  It would be great for other developers on my team to benefit from shortcuts I've created for myself, and vice versa.</p>

I know this post is very old, but I found another good way (IMO) for sharing snippets, maybe in addition to the solution @Kevin Dente provided:
using git
!!
I shared successfully snippets and project templates creating an ad hoc git repository named
snippets.git
, I cloned it into the
MyDocuments\Visual Studio 2015\Code Snippets
folder and now I can add, edit or delete snippets from
MyDocuments\Visual Studio 2015\Code Snippets\Visual C#\My Code Snippets
(or
Visual Basic
folder), then commit and push to make them available to all other team members with the same configuration.
The principle is applicable also to the project templates, you can create a
templates.git
repository and clone it to the
MyDocuments\Visual Studio 2015\Templates
folder, then create the templates under the
MyDocuments\Visual Studio 2015\Templates\ProjectTemplates\Visual C#
folder (or
Visual Basic
folder).
This way the developers can periodically pull the two repositories and have up to date snippets and templates, available almost everywhere the git server is reachable (anyone said github? or bitbucket?).
Of course I am using Visual Studio 2015, but the method can be used with previous and, I think, future versions of Visual Studio.