> How much of the awarded money does the plaintiff actually receive as compensation in such a case? Do they have to share some of the awarded money with the lawyers?
Depends on the agreement with the lawyers. The two main payment terms are:-
A) you pay your laywer their fees no matter the outcome. In this case you keep all the money awarded to you.
B) A contingency agreement. In this case you don't pay the laywer directly for their fee's however payment for your attorney's services is "contingent upon" your receiving some amount of compensation and the award is split based on the agreement. NOTE: Some types of cases can not be based on contingency for iirc ethical rules reasons.
It _COULD_ be a fee based contingency agreement saying "If we win, you will pay all our fees", but the more common agreement is the award sharing agreement saying for example "If we win you agree to pay 33% of the award given".
> are they also entitled to the punitive part of the damages?
Yes, however often punitive awards are often capped which the jurors are not told about. So lets say X state caps punitive awards to 1 Million, but the jury gives 10 million, the punitive award is reduced to the 1 million cap.