Take a look at Van Proeyen's online notes on SUGRA, or his book with Freedman. There it is really explicitly explained, both of-shell and on-shell :)
The Vielbein's DOF's are easy (of-shell): it's a matrix, so it has D^2 components in D dimensions. Local Lorentz transformations make you subtract 1/2*D*(D-1) components from it, leaving you with 1/2*D*(D+1) components for the metric. Which is the right amount for a symmetric tensor like the metric.