Take $n$ copies of a topological space $M$, and consider this as a single (disconnected) topological space $X$. Let $f:X\to M$ be an $n$-to-1 map that maps the $n$ copies, via separate identity maps, onto a single copy $M$.
Now consider the pushforward or direct image map $f_*$. Suppose we place a different line bundle $L_i$ ($i=1,\ldots,n$) on each copy of $M$, and we map this by the pushforward to give an object $V$ on $M$.
My question is whether the resulting object $V$ is a direct sum of the line bundles $L_i$, i.e. whether $V=\bigoplus_{i=1}^n L_i$, or if the result is a more general vector bundle (or perhaps it is not a vector bundle at all).
As suggested above by Nefertiti, we can study, given $U \subset M$, what is $f^{-1}(U)$ and what that tells about the pushforward sheaf $V$. Well $f^{-1}(U) = \coprod_i U_i$, where $U_i$ are disjoint copies of $U$ in the various disjoint copies of $M$. So by sheaf properties, $$ V(U) = F(f^{-1}(U)) = \bigoplus_i F(U_i) = \bigoplus_i L_i(U_i). $$ This holds for any open set so the pushforward sheaf $V$ is the direct sum of the $L_i$.