As noted above, make sure these issues get worked out before the season starts. Everyone involved in injury management needs to be on the same page. At every hockey game, our program has an ATC, 2 student-trainer/EMTs, 2 paramedics, a GP, Ortho, and Dentist. The visiting team as an ATC and sometimes a doc as well. They all know their roles during injury management, and we've yet to have a situation with serious headbutting with the EMS side (the Doctors can be a bit different since there are many of them and they are not working in their usual environment).
At our program, the trainer is "in-charge" at all times, unless the injured player, coach, or official has sustained a life threatening injury. We are lucky in that we have the same paramedic crew (or at least one of them) at every game, so we do not have to hash it out with a new crew weekend.
The paramedics are an essential part of our medical team. They help us with our c-spine protocol on the ice, provide supplies and equipment that we don't generally have access too, and manage the life threatening injuries that our Sports Medicine staff don't have experience dealing with. It's also great to have someone able to start a line and give zofran to a player that's been battling illness and is dehydrated and nauseous before the game.