Bennett tries to talk most of them out of the idea but, like many physicians, is torn between the need to protect individual patients and his duty to the community.
"I have fielded 15 or 20 calls myself," Bennett said.
The H5N1 avian flu has only infected just over 120 people, but it has killed half of them and the world's top health experts have been increasingly vocal in their warnings that it could become a deadly pandemic.
The virus is spreading among flocks of poultry and every new outbreak makes headlines. It has moved into Europe and politicians have been making loud pronouncements about how poorly the world is prepared to handle it.
Add to that the lesson demonstrated by Hurricane Katrina. When the storm hit the US Gulf states of Louisiana, Mississippi, Florida and Alabama at the end of August, residents found they were on their own for days and weeks, with little or no federal help.
So the highly educated and motivated residents of north-west Washington want to take matters into their own hands. They have heard about Tamiflu, a drug used to treat seasonal flu and one that countries are stockpiling to use against H5N1 should it cause a pandemic.
They want some, too.
"You can understand where people are coming from," Bennett said. "We had the same thing after anthrax."
In 2001, someone mailed a series of anthrax laced-letters to addresses in New York, Washington and Florida. Five people died, including two postal workers in Washington, and many people clamoured for the antibiotics that could prevent anthrax disease.
The same demand may be developing with Tamiflu, which comes in a pill, and, to a lesser degree, Relenza, a similar drug that is inhaled and so not suitable for use by people with asthma.
While pharmacists in Hong Kong report a lively demand, the drugs are unavailable in Indonesia and Cambodia, hard hit by avian influenza, and some other Asian nations because authorities there did not place orders for them.
Roche and Gilead Sciences' Tamiflu and GlaxoSmithKline's Relenza can prevent and treat influenza if used very quickly. But both are in short supply.
"We are not recommending hoarding," Bennett said although he added some colleagues disagree.
"If people are hoarding it, my main concern is that there won't be Tamiflu for the people who need it," Bennett said.
Dr Anne Moscona, a professor of paediatrics and microbiology at Cornell Weill Medical Center in New York, said she is trying to set an example.
"I am not stockpiling. My infectious disease colleagues are not stockpiling. Premature use, incorrect use and excessive use of these drugs will lead to us losing them when we really need them. We need to encourage our government to make and stockpile enough to treat everybody."
Bennett is more worried about seasonal influenza, and wants his patients most at risk to have treatment available should they become infected. A run on Tamiflu could mean pharmacies will not have stocks when they are most needed at the height of the annual flu season.
"I have written five or six prescriptions and the people I have written them for are patients with chronic diseases, such as patients with asthma," he said. They are directed not to take the drug without calling Bennett first.
Doctors agree it would be unwise for anyone to take Tamiflu unless diagnosed, definitively, with influenza. When used against seasonal flu within the first day or two of infection, it can reduce the severity of the disease.
But some tests suggest higher doses might be needed with H5N1, depending on what kind of mutant strain eventually emerges.
"No one really knows what the dose would be, how long you would have to take it," Bennett said.
"If bird flu comes to the community, it is not going to be gone in a week. If you want Tamiflu in the house, you will need it for every person and you will need 15 or 20 prescriptions."
Dr D.A. Henderson, a smallpox expert who advises the federal government and who helped set up the Center for Biosecurity at the University of Pittsburgh, agrees.
"I personally would not advise trying to take it as a preventive because if there is an outbreak in your city ... that is going to go on for six to eight weeks," Henderson said.
And eventually, the virus will change and become resistant to the drug. This has already happened with an older flu drug, amantadine, and one Vietnamese girl treated with Tamiflu grew a drug-resistant strain in her body by the time she recovered.
"Tamiflu being what it is and flu virus being what it is, there are going to be a lot of people out there taking Tamiflu," Henderson said.
"Certainly people are going to be getting resistant strains and they will be passing it along."
If a person happens to be among the first infected and treated, they personally will not be at risk, Henderson said.
"You are going to be getting H5N1 and you going to be developing antibodies to it. You are not going to have a problem because you'll have the antibodies. But if you pass it on to somebody else you'll pass on a resistant strain."