This scenario above has played out several times when I've asked the question. In reality they don't have a Mutual Aid Agreement, nor a Memorandum of Agreement (MOA), nor a Memorandum of Understanding (MOU). They do have another type of agreement also called an MOU that stands for a "Memory of Understanding."
While I'm a big "Trust" guy and am more than willing to do a handshake deal with an individual, it is not a good procedure for any public or private organization to operate on verbal agreements when it comes to disasters.
Disasters by their nature can be dangerous events. What if someone assisting in another jurisdiction is seriously injured or killed? Who is responsible? If the vehicle deploying to the scene as a mutual aid asset hits and kills a pedestrian, who is liable?
A quote I've used often is, "The faintest ink is more lasting than the most powerful memory." Call it a mutual aid agreement, MOU, MOA or whatever--it should be in writing.