I created some connection templates using the RAM Connection mode inside STAAD.Pro. How can I get these templates transferred from my machine to a second user's machine?

To begin with I would direct you to the wiki article which you may be already familiar with as my suggestions would be based on that

http://communities.bentley.com/products/structural/structural_analysis___design/b/analysis_and_design_blog/archive/2010/12/09/how-to-customize-a-ram-connection-template-in-staad-pro

Specifically refer to the section titled Your Custom Template File

First you need to copy the <group.name>.con file from within the folder in your machine as indicated in that section 

Please note that the file may be at the following location as well depending on which version of STAAD.Pro you are using
 
C:\Users\<user.name>\AppData\Roaming\Bentley\Engineering\STAAD.Pro V8i SS5\RAM Connection\Database\Connections\User

Copy the file and paste it at the same location within the second user's computer where you would like to copy the connection templates.

Now copy the same .con file in the following folder of the second user's machine

C:\…\SProV8i\STAAD\RamConnection\Database\Connections\United States\

From your machine, copy the connections.ini file from within your C:\…\SProV8i\STAAD\RamConnection\ConnectionTemplates\ and replace the same file in the second user's machine with that.

It is always better to keep a backup of all the files you are replacing in the second user's mchine, before you actually replace these.

This should get your templates transported to the second user's machine.