You can also drive a friends car if they have sentri too. I didn't know this. I was told that if it has US plates I can drive. If it has MX plates I can ride. I imagine it is the opposite for MX citizens.
The rules are the same in regards to crossing the border in either direction. If you are driving across the border, you need to be a legal permanent resident or citizen that matches with the plates on your car. Keep in mind this is in regards to crossing the border. I don't think there is an issue with driving in the country in either case as long as you have a drivers license.
Mexican citizen or permanent resident - must be driving car with Mexican plates when crossing the border
US citizen or permanent resident - must be driving car with US plates when crossing the border
Not sure what applies for dual citizenship. Each country requires that you declare your citizenship if you are a citizen of that country. So if you a 'dual citizen' of Mexican and the United States this could be an issue. When you enter the US you need to declare that you are a US Citizen. When you enter Mexico you need to declare the you are a Mexican citizen. I don't know that your single car can have plates from both countries (although I have seen trucks with both plates).
Maybe someone else who has a better legal understanding of this can clarify.