"$X$ follows $Poisson(\theta)$ and an i.i.d. sample $X_1,...,X_n$ has been drawn"? What does "$X$ follows $Poisson(\theta)$ and an i.i.d. sample $X_1,...,X_n$ has been drawn" mean?
Does $X$ follows $Poisson(\theta)$ $\implies$ $(X_1,...,X_n) \sim Poisson(\theta)$

or

Does $X$ follows $Poisson(\theta)$ $\implies$ $X_i \sim Poisson(\theta)$

i.e. what is $X$ referring to?
 A: $X$ is the random variable that relates, as a function, the possible outcomes $\Omega$ to the sample space, $S$. In the case of a Poisson-distributed random variable, $S\in \mathbb{N}$: the possible realizations of $X$ span from $\{0,1,2,\cdots,\infty\}$ with different probabilities, governed by the pmf, $\frac{\lambda^k\,e^{-\lambda}}{k!}$. When you draw samples from such a random variable, you actualize its conceivable values into random variates, which are your $X_i$ values. From Wikipedia: "The realizations of a random variable, that is, the results of randomly choosing values according to the variable's probability distribution function, are called random variates."
Therefore it is the random variable $X$ that follows a Poisson distribution, $X \sim \text{Pois}(\lambda)$, and not the actualizations (or realiztions) - the random "variates", $X_i$, which are actual integers (typically number of events or counts), without any uncertainty associated to them. 
Draws from a variable $X \sim \text{Pois}(4)$ can be plotted on a histogram as individual realized points, and the overall distribution becomes apparent as the sample increases:

