They ask you for identifying info when you call them for service. Usually name and phone number is all. I have asked the agents once or twice if I need to present my card or other id to the tow truck operator, and have been told "no". They tell them who you are, where you are, what you're driving, and what the problem is before they arrive at your location. It's never been an issue for me. I've asked the tow driver if they need to see my card, and thay also say "nope".
That said, I like the plastic cards, and have kept some to carry, just in case. :W