The $p$-th guest receives a share of 
$$s_{p,N}=p N^{-p-1} (N-p+1) \frac{\Gamma(N+1)}{\Gamma(N-p+2)}$$
For large $N$ I substitute the asymptotic expansion of the Gamma function,
$$\Gamma(z)\approx e^{-z}z^z(2\pi/z)^{1/2},$$ 
to arrive at the approximation
$$s_{p,N}=p(n-p+1) e^{1-p} \sqrt{\frac{1}{n+1}}\sqrt{\frac{1}{n-p+2}} (n+1)^{n+1}  n^{-p-1}  (n-p+2)^{-n+p-1}$$
which agrees very well (compare the dots with the continuous curve in the plot below, for $N=1000$)

<IMG SRC="https://ilorentz.org/beenakker/MO/pochhammer.png"/>