$\textbf{Answer}$

The probability of having at least two people sharing the same birthday exceeds $50\%$ among $\boxed{23}$ people, and exceeds $99.9\%$ among $\boxed{70}$ people.

$\textbf{Analysis}$

There are totally $366$ different possible birthdays (taking leap year into consideration). Then, the probability for $N (\le 366)$ people do $\underline{not}$ share the same birthday equals $$\frac{366}{366}\times\frac{365}{366}\times\frac{364}{366}\times\cdots\times\frac{366-N+1}{366}=\frac{366!}{(366-N)!\times 366^N}$$

This is because the first person has $366$ choices without collision, the second person has $365$ choices, and so on. Accordingly, the probability of having birthday collision is $$1-\frac{366!}{(366-N)!\times 366^N}$$

Setting $N=23$ in the above formula yields $50.63\%$.

Setting $N=70$ in the above formula yields $99.91\%$.

$\textbf{Note}$

These two numbers, $23$ and $70$, are well-known among people who are interested in probability.