Responding to the PD is very similar to a standard takeout double except that opener always has a good hand which makes bidding easier for responder.
Free bids tend to limit hands in that they contain less than invitational values; any jump would show invitational values as in standard although we invite more frequently with distributional hands.
A cue-bid is forcing. Both responder and the PDer follow up with natural bids allowing room to find the best spot. We play South African Transfers: a transfer to four of a major by bidding the corresponding minor, C => H & D => S, and this allows the PDer to super accept by bidding the step in between.
The ranges for responding in no trump are slightly adjusted as compared to standard: 1 NT = a good 4 to a bad 7, 2 NT = a good 7 to a bad 9, 3 NT = 10-14, etc. This works out well because many times we right side the no trump, forcing the strong hand to be on lead, resulting in an overtrick. Notice that we also avoid the danger of overcalling a no trump and getting doubled off for penalties.
A definite advantage to the PD comes from the ability to penalize the opponents at a low level, especially if we're white and they're RED.
For example, if holding S Kxx H Axx D QJ9x C xxx and partner made a PD over LHO's 1 diamond opening, we would float the double. We'll make 3 NT, but LHO won't enjoy playing this one out of his hand. Floating the double is probably right on two types of hands. The type of hand given in the example that's all primed, balanced with a couple trump tricks or a hand with just LHO's suit and no side suit precluding your side from having a big fit.
These aggressive passes have been successful; a further benefit develops from passing these doubles. If opener runs from the double, it's a reasonable possibility that their bid was psychic and partner would now play any of your bids in that suit as natural. It becomes very difficult for the opponents to dink around as we can draw so many more inferences from an auction than in standard.