I have this same problem. I contacted SW support to find out if a request type can be used with multiple tech groups.
The short answer is no.
The long answer is, in the current version of WHD you can only have one request type associated to one tech group.
So lets say that you make a tech group called "network" and you assign specific request types to this tech group to handle. Then you make another tech group called "support" and assign some request types to this tech group. Then you realize that support tech group needs to handle one of the same request types that has been assigned to the network tech group and you try and add that request type to the tech group that will also start handling the same request types. You will find that you can't add it to the tech group.
I have heard that this may or may not be fixed in the next release.
Hope this helps.
You have the option of restricting your Techs to ONLY see tickets for their own Tech Groups. This is done by assigning a Tech Permissions set to the Tech(s). Within the definition of Tech Permissions, the specific setting you want to toggle on is called "Limit to Assigned Tech Groups"
So, if you have a Request Type (example: HR Info Update)
and that Request Type is assigned to a Tech Group (example: HR Techs)
you could restrict those Techs that belong to HR Techs by turning that permission on for them.