Hi Sue,
From my experience, billing is driven by the fee earners/secretaries in the small to medium sized firms. That said, I am hearing of more and more medium firms moving towards centralised billing. In the larger firms, it tends to be a designated billing team.
What practice management system are you currently using? Is the intention to completely remove the fee-earner from the process?
I will pose this question for you on our next Social Support Team meeting on the 26th of this month to gage what others are currently doing.