Some people are afraid of the radiation they put out.
Another thing is intrusion into monitoring what one uses. I'm not talking just a reading like the old meter, but constant communication back to the power utilities constantly, and information about the electric user.
The advantage (to the power companies) is that your power can be turned on or off remotely without sending a guy out.
I'm not afraid of #1, despite my interest in health. I think there's a lot more radiation coming from our electronic gizmos elsewhere. As for #2, that's a consequence of technology. Already the cable and Internet companies can tell a lot more about a person than a smart meter could, so gathering personal information is a moot point since this only adds a small amount of information relatively speaking. This only becomes a problem when this information is used wrongly. For example, saying to a paying customer "Hey, the government saw your records and told us utility people to tell you to stop using so much power in order to keep the earth green."

Now that's invading personal space. But if the power companies need to know where the power is consumed for proper infrastructure, that is another thing.
On a similar example, some people (due to privacy issues) protest cameras on the street. I don't because when one goes out in public they are in plain view of anyone on the road anyway. What's the difference if a COP on the beat saw what you did instead of a camera (as long as the technological limitations are considered if used for evidence)? Now, if the camera could have x-ray vision into your car (or thru your clothes) :o, that would be unconstitutional because a person has the right to be secure in their persons, etc because unlike the street, that is personal space.
As a conspiracy theorist myself, I am more concerned about the taking away of constitutional rights, and the big stuff. Sometimes people stress over little token stuff that's really nebulous.