Which binds first, product or factorial?
This would depend on the author, but the former notation would be much more common: $$\prod_{i \in I}i! = \prod_{i \in I}(i!)$$
If the product itself was factorialized, it would most likely be written as the latter: $$\Bigg(\prod_{i \in I}i\Bigg)!$$
edit: added the bolded word much.
The convention \begin{align*} \prod_{i \in I}i! = \prod_{i \in I}(i!)\tag{1} \end{align*} is also affirmed by the operator precedence rules stated in OEIS.
For standard arithmetic, operator precedence is as follows:
Parenthesization,
Factorial,
Exponentiation,
Multiplication and division,
Addition and subtraction.
and since the product sign $\prod$ is just a short-hand for successively using the multiplication operator, the convention (1) is valid.
I would see it as $$\prod_{i \in I}i! = \prod_{i \in I}(i!)$$
Like the $\sum _i a_i^2$ which is $\sum _i (a_i^2)$ not $(\sum _i a_i)^2$