How to know whether a contact form is only defined locally or globally? As described e.g. here the following is the standard contact form on $\mathbb R^{2n+1}$:
$$ \omega = dz + \sum_{k=1}^n x_k dy_k$$
Similarly, the following is the standard contact form on $S^{2n+1}$:
$$ \beta = \sum_{k=1}^{n+1} x_k dy_k - y_k dx_k$$
These are two examples of contact forms. Some contact forms are defined globally, everywhere on a given manifold. Some are not. At least that's what I understand so far. 
My question is:

How can I determine whether a given contact form on a manifold is
  globally defined on that manifold or only locally?

 A: May be here you are doing some confusion. 
In the first case you are allowed to define a global 1-form in $\textit{local}$ coordinates just because you are in a lucky case in which you can cover your manifold by a single coordinate chart.
In the second case the form $\beta$ is defined on $\mathbb{R}^{2n+2}$, and in order to get a 1-form over $S^{2n+1}$ (that you will denote again with $\beta$ abusing of notation) you are implicitly taking the pull-back of $\beta$  via the inclusion map $i: S^{2n+1} \to \mathbb{R}^{2n+2}$.
Another connected question (that may be was your original one) could be: when can I define a contact structure by just using a $\textit{global}$ 1-form?
The answer is easy and does $\textit{not}$ involve the contact condition. You can do it if and only if the hyperplane field is $\textit{coorientable}$, that is its normal bundle is trivial (i.e. has a $\textit{global}$ non-vanishing section). In fact, you can easily check that a 1-form defining  a hyperplane field induces a $\textit{global}$ non-vanishing section of its normal line bundle and viceversa.  
