I'd always assumed that the Emporer shaved off the top(or bottom) of the personnel lists(the mindlessly loyal of course) gradually or used slave labor to have enough workers to build the first Death Star.
It could have been easy then to keep it secret from the senate (its a big galaxy after all).
Once the senate was dissolved, he brought in all of the manpower he could muster from all over the Empire to build the second, which resulted in it being completed much faster.