It isn't carbon dating that is used for dinosaurs. It's meaningless for anything over about 50,000 years. Other methods are used.
There is a possibility of dinosaurs living alongside humans though. Just because dinosaurs lived a long, long time ago doesn't mandate that they can't also live now.
Of course, with one class of exceptions, not finding them now suggests that they aren't living alongside humans.
The exception is the birds which are classed as dinosauria so they do in fact live along side us which I think is very cool.