Check the representation agreement that you signed with them.
When I work with sellers and take listings I explain transaction coordination up front and even write it into the listing agreement. That way it is fully explained what the coordinator does, what the cost is, and when the fee will actually get paid. For buyers the same is true. If I do choose to have the buyer pay for coordination, it gets written into the buyers representation agreeement or into the Residential Purchase agreement, explained and everyone is on the same page.
Of course if you are using a “full service” agent that is getting 2-3% commission and not giving you any rebate, it goes without saying that they should pay.