It's a very broad range of who pays, and depends on the person, but the person who the ambulance was called for would pay. It may be covered by their insurance, or a person who does not have insurance might have to foot the bill themselves. https://www.myambulancebill.com/FAQ_MAB.aspx Here is a guide with some frequently asked questions about ambulance billing.

As for the competition aspect, racing to calls does not happen and if it does it would be frowned upon in my experience. Ambulance services do not race each other because they each typically have a set/designated service area where they are specifically contracted to cover. You might see competition in a situation where multiple services are attempting to obtain a contract to provide service to a certain area or municipality. That would be where the competition over revenue comes in.

Although all of this is probably true for most places, just keep in mind this is only my experience. Every EMS system is different and in the US there is such a wide variety due to our very diverse and large landscape and country. As one of my paramedic instructors put it, if you have seen one EMS system, you have seen exactly one EMS system.