Obviously I'm late to this thread, but in my experience it usually works like this: he would offer a draw on his time (during his move), then make his move, and then you can use as much of your time as you want to think it over and decide whether or not to accept.
Offering draw during move.
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A player can offer a draw on their time, however youre within your right to ask them to make their move.
You have as long as you want, a draw offer may not be retracted, and is only rendered invalid upon declining it, or making your move.
As I recall, the proper ettiquette is to make your move, then offer a draw, then press the clock. The opposing player can then think it over (on his own time), and either accept, refuse, or make a move on the board (which counts as a refusal).
He can offer a draw on his move, but you can choose to answer either before he moves, or later, during your move, so it's probably not to his benefit to commit to a draw offer so early.
The case described by dbag301 is quite interesting. Player A makes a draw offer on his move. Player B is in his right to to ask A to make his move. However instead he can just start thinking or make it look as if he was considering a draw offer. Sooner or later player A will realise that he is at risk of loosing on time so he might make his move. But a draw offer is still pending! So now player B can decide if he accepts it or tries to make a profit from his opponent zeitnot.
That almost resembles a dilemma of telling or not to your opponent that he forgot to switch the clock after making a move.
I believe the rules say that you have to make your move, hit your clock and then offer a draw. At the last OTB tournament, my opponent offered a draw during his move. He had not made a move yet and asked for a draw. Looking back at it, I realized that it took me a little while to think about the offer before I accepted it. The thing is that I was using his time to think about accepting it or not.
How long does a player have to accept or decline a draw after it is offered? If your opponent asks for a draw during his move (and has 5 minutes left on his clock), could you just think for 5 minutes and let his time run out? What about 30 minutes? Or would he be able to make his move during that time and shift the time to you?