Often when that call goes out over the public address system on a flight, it's an indication that something is very wrong. Recent news reports about the 10 year old who died during a flight and the serious heart ailment suffered by Carrie Fisher have brought this to everyone's attention but the reality is very different from what you might have thought.
True medical emergencies are thankfully very rare on flights. A study found that they happened in approximately 1 out of 604 flights involving about 44,000 passengers out of the 2.75 billion who were travelling so you can see that the numbers are actually relatively low (although you could argue that even one is one too many and I wouldn't dispute that at all).
Most calls were for help dealing with someone feeling faint or lightheaded. Next most often were breathing difficulties and then nausea, followed by heart issues and seizures. Cardiac arrests (where the heart actually stops beating) were responsible for less than 1% of onboard medical emergencies but were responsible for the vast majority of deaths happening during a flight. Of the 36 people who died during a flight in this study, 31 died as a result of a cardiac arrest.
Flights involving US carriers have crews trained in first aid. Furthermore most international airlines have an arrangement so that there is a doctor experienced in dealing with in-flight emergencies available by radio for the air crew to consult with. Similarly most airlines stock medical kits but what the kits contain can vary significantly from airline to airline.
No health care provider, as far as I know, has ever been sued as a result of their attempts to help but I do know that almost every medical association warns physicians not to volunteer if they are not at their best, for example after consuming alcohol. On the other hand, with the exception of Australia and Europe, nowhere is a health care professional under a legal obligation to help out.
Contrary to what you might expect, long gone are the days of special perks for the volunteer. My own personal experience is a perfect example. On a very long, multi-stage trip from a medical conference where I was lecturing, I answered the call and took care of an ailing passenger. Missed my meals, the inflight entertainment and any sleep. When we landed and the passenger was taken away in an ambulance, I asked the airline if they would move my wife and me from economy to business for the rest of the flight. Don't you know they refused! That has been the experience of my colleagues in similar situations as well.