I am not a lawyer, and this is not official legal advice. In addition, laws vary between countries, so the situation can always be different for your country or the country you're diving in.
I've heard dive professionals (divemasters and above) talking, in person and on places like ScubaBoard, about when they are diving on vacation (as opposed to diving while working). There seems to be a common sentiment: when the shop wants to see my C-card, only show them my open water or advanced open water card. The less they know the better.
There are two reasons to do this. One is perfectly valid, while the other is unfounded.
I don't want to be a dive leader
If a dive professional is on vacation, then they want to be on vacation. They don't want to be responsible for other divers, even implicitly.
Now, no shop is going to ask a vacationing diver to help out with conducting classes, but it is possible that by knowing you're there they (unconsciously) modify their behavior, shifting responsibility onto you. For instance, if grouping divers for something like a drift dive, they may arrange differently when taking your knowledge and experience into account. This could implicitly put more responsibility on you.
This is somewhat valid, since divemasters make judgments all the time based on the experience of the group they're diving with. So you shouldn't lie about your experience, but if a line is crossed where you no longer feel solely responsible for yourself then the vacation is over.
Another thought is that you may be expected to help in the event of an emergency. After all, you're trained to handle these things. Some people don't want this unexpected responsibility.
This is a valid sentiment, and so it might be reasonable to only flash your advanced open water card when asked. However, there is no reason to lie when asked about your diving experience (number of dives, types of dives, etc.). By doing that you could unwittingly put divers who are actually less experienced in a less secure position during the dive.
I don't want to get sued
Another cited reason is that if it is known you are trained and certified to handle diving emergencies, any lack of action on your part in the event of an accident could be cause for litigation.
Let me state right now that this is completely untrue.
I've talked with a friend who is a lawyer, and he tells me there is no precedence for anything like this. It is true that they could file suit against you, but I could file against anyone for anything if I really wanted. The suit would be promptly thrown out, and any worthwhile lawyer the person involved in the accident consulted should advise against filing a hopeless case.
There are two cases where you could be sued. The first is if you are a shop employee and an accident occurs. This is why you have professional liability insurance. The second is if something happens, you decide to help, and you really screw it up by not following established emergency protocol given the situation. There are Good Samaritan laws in place to protect you, so as long as you follow accepted procedures for accident management and don't attempt anything you haven't been trained for you should be fine.
I wanted to clear up the misconception about getting sued for not helping. We could certainly debate the existence of a moral obligation to help, but we are strictly talking about the law.
One last concern is if your professional diving organization (PADI, NAUI, etc.) has any statutes for members' behavior in the event of an emergency. For instance, a medical doctor that refuses to help a victim could face penalties from medical associations of which they are a member. Whether this is the case for pro divers, I don't know.
If you have any further insight, let everyone know in the comments.