In short, yes I probably would, but for a bunch of different reasons.
First, our department gets paged out, and I always have my pager with me, (if they need a second crew, I will be the one to respond 9 times out of 10). Even if we don't have our pager on, we get all of our calls text messaged to our cell phones. We live in a town of 1200, and have 11 people working EMS. If the page is for a man down not breathing, and I walk in wearing jeans and a tee shirt, and the patient says "no thanks I will wait for the guys in the ambulance" I will gladly leave him alone. Otherwise I am sure said patient or bystanders will be happy I responded
Our department runs and combo of basic crews or ALS crews. So the responding crew might be BLS, and chances are they will be calling me to help anyways, cause 75% of the time I am the only ALS available. If the man is not breathing, the call will require more people than the driver and EMT that are responding. They would be very happy to see another EMT already on scene, even if the person is wearing jeans and a tee-shirt. Our uniform is pretty much jeans and a tee-shirt anyways, it does have our department name on it, but jeans and t-shirt just the same.
Our responding crew has 5 minutes to get to the ambulance, then however long it takes to get on scene. There is alot I can do in those 5-10 minutes that will speed up patient care. Begin CPR if it hasnt been started, get history, or interview bystanders, make sure the area is clear for the gurney, ect.
One thing to keep in mind, if you do render aid, YOU may have to transport. We recently had a call for a lady in anaphylaxis, the responding crew was BLS, but had called for ALS mutual aid. They got on scene, and one of the bystanders was a Dr, she was screaming that the patient needed EPI (she was freaking out), the crew explained they couldn't give EPI, but would monitor the airway, load the patient ect, and another unit was responding to give epi. She said well I am a Dr. give me the epi and I will give it to her. The crew said that would be fine, but then SHE would have to ride to the hospital with them. She changed her mind lol. A medic from a neighboring agency walked up and gave the patient the epi, and was willing to ride in, but the ALS rig had arrived so he didn't need to.
So remember if you respond to a call just to be helpful, you still have to give that patient to someone that is at your level of care or higher!