Product forcing and generic objects If we start with a model of $\sf ZFC$, $M$ and $(P,\le)\in M$ is a notion of forcing, $G\subseteq P$ a generic filter, then in $M[G]$ we can define some generic object from $G$. For example, if $P$ is the Levy collapse of $\omega_1$ to $\omega$ then $G$ defines a new function $f\colon\omega\to\omega_1$ which is bijective. 
Now suppose that we have a product forcing $P=\prod P_i$ in $M$, then the generic filter $G$ can be projected on every coordinate and $G_i$ (its projection) is a generic filter over $P_i$, which defines some generic object. Then a priori we can think that $G$ defines some generic collection $\{g_i\}$ such that $g_i$ is the generic object defined by $G_i$.
So for example, if we take the product of two Cohen-like forcings, one adding a subset of $\omega$ and the other adding a subset of $\omega_1$ - we can think of the collection as the pair of the new subsets.
In Jech Set Theory, 3rd Millennium edition, in the relevant chapter (Ch. 15) Jech discusses this very shortly, proving some basic theorems about this. However in the exercises there is only one problem related to this issue:

Let $P$ be the notion of forcing (15.1) that adjoins $\kappa$ Cohen reals. Then $P$ is (isomorphic to) the product of $\kappa$ copies of the forcing for adding a single Cohen real (Example 14.2).

This means, that we can think of the product of $\kappa$ Cohen forcings as adding $\{g_i\mid i<\kappa\}$ as a set of $\kappa$ new Cohen reals, just like we would think at first.
However, there is no mention of this being true or false in a general framework. So to my question:

Suppose $P=\prod P_i$ is the product of $\kappa$ copies of some $P'$ a fixed notion of forcing, can we automatically assume that $G\subseteq P$, a generic filter, adds a set of $\kappa$ new generic elements, each defined by a generic filter, $G_i$ over $P'$?

If this is true, then we can ask even further:

Suppose $P=\prod P_i$ is a product of $\kappa$ notions of forcings, can we say that $G\subseteq P$, a generic filter, adds a set of generic objects each defined solely by $G_i$?

 A: For your first question, yes, it is true in complete generality. If $G$ is a $V$-generic filter on the product $\Pi_i P_i$, then the projection of $G$ onto each factor, that is, the set $G_j$ consisting of the $j^{\rm th}$ coordinates of the conditions in $G$, is a $V$-generic filter for $P_j$. This is because if $D\subset P_j$ is any dense subset of $P_j$ in $V$, then the set of conditions $p\in \Pi_i P_i$ that have their $j^{\rm th}$ coordinate in $D$ is dense in the product forcing, and thus it is met by $G$, and so $G_j$ meets $D$. 
In particular, if the product consists of $\kappa$ many copies of a single nontrivial forcing notion $P'$, then the product forcing will add $\kappa$ many $V$-generic filters for $P'$. If $P'$ is nontrivial in the sense that there are incompatible conditions below any given condition (that it, it is splitting), then it is dense in the product that the generic filters $G_j$ added on each factor are distinct, since for any pair $i,j$ the set of conditions in the product for which the $i^{\rm th}$ coordinate is incompatible with the $j^{\rm th}$ coordinate is dense in the product.
Conversely, we can reconstruct the full product generic $G$ from the projections $G_j$, since a condition is in $G$ if and only if its projection on coordinate $j$ is in $G_j$ for every $j$. Perhaps this is the what you ask in your second question? 
But not every collection of $V$-generic filters $G_j$ for $P_j$ will give rise to a generic for the product forcing. For example, we can't have them be all the same on every coordinate for the reasons mentioned above. The additional property that the factor filters form a generic filter for the product is called mutual genericity. 
