Yes, the employee. Why do you think it's shady? In my years of working in and around people with the same setup I've never heard of any complaints, and I know most of the other large consulting companies do it the same way.

AFAIK the thought behind it is that not every expense on the card is for work - as a consultant, I travel every week and I put all kinds of expenses on the card (not all of which are strictly "for work" or allowed to be reimbursed by the company). This way, my company still pays the bill for whatever expenses I mark as reimbursable, but the company doesn't have to worry about being on the hook for when I use my card to buy a pair of pants.

To me, it's basically the same as having a personal CC and my company reimburses me for the expenses - except in this case, it's a lot easier for my company to pay part of the bill directly (instead of me being an intermediary) and I also don't have to worry about hitting a credit limit.