When I was an EMT, I always had my wallet card in my possession. If someone of authority (such as a police officer) were to request it, I would show it. Would I show it to a 15-year old Boy Scout? Maybe - but I'd put the odds at 90/10 against. Unless the patient was stabilized, I doubt I would take the time to ask for the ID of anyone coming up with an offer to help.
Interesting question on Duty to Help - and the answer is a bit complicated.
When it comes to EMS personnel, in general, there is no Duty to Rescue/Help/Respond beyond some state's statutory duty of it's citizens duty to help, IF the EMS member is off the clock and not being paid. They only have a duty to respond if it is part of their employment and they are being paid for the response.
An On-duty Paramedic cannot refuse to provide aid to anyone - because s/he is on duty and therefore being paid, s/he must respond and must act. Once s/he punches out though (goes off duty), s/he no longer has a legal duty to act, even if they are still in uniform. For example, if a Paramedic is on duty and shopping for groceries and a person collapses in from of them, the Paramedic must provide aid. However, if that same Paramedic is off duty (either in or out of uniform) shopping for groceries for the family, and someone collapses, the Paramedic does not have to render aid (most will so lets not get bogged down in a discussion of whether they will or not - this is about must they do so).
Here's the key question relevant to EMS Duty to Act - Is the person being paid - any amount - at the time of the act. If they are not being paid at the time of the act - they have no duty to act. What this means legally is that if you are a volunteer, you have no legal duty to act, even if you are "on duty". In a rural area that only has a volunteer EMS ambulance service? As long as those volunteers aren't paid - they have no legal duty to respond and can, in fact, just not show up if they all decide they don't like you. That's why a lot of volunteer EMS services operates as paid--on-call services - it gives them a duty to act.
Now for the complication: - there are some states that also have citizen duty to act laws that require citizens to render assistance to victims. So in those states, an EMS member would be required to stop and provide aid. EXCEPT - they wouldn't be required to provide aid related to their licensing even under most of these laws. The Citizen Duty to Act laws are very basic - generally, you just have to stop, call for help and wait until help arrives and that's it - you don't actually have to give CPR or stop someone from bleeding - you just can't abandon them.