In my current company, my team is part of our client services team. We generate training services (paid and free) for our clients. Also alongside my team on the overall services team is our Implementation team and Consultation team. This is probably the best place for what my team provides. While we do not have client-facing trainers (on my team), we still provide several services.
The good thing is that Training teams come in all shapes and sizes and they contribute to the organizational mission in a variety of ways. There are some teams who are part of the IT team because they can focus on development. Some are part of the Product team because they focus solely on launch. Many are part of HR because they focus on organizational development.
If you are part of a training team, where is your team positioned?