  Given two identical linear codes $\mathcal C$ over $\mathbb F_q$ of length
$n$, we independently pick one codeword from each codebook uniformly at random.
A $\textit{sumset}$ is formed by adding these two codewords entry-wise as
integer vectors and a sumset is called $\textit{typical}$, if the sum falls
inside this set with high probability. We ask the question: how large is the
typical sumset for most codes? In this paper we characterize the asymptotic
size of such typical sumset. We show that when the rate $R$ of the linear code
is below a certain threshold $D$, the typical sumset size is roughly $|\mathcal
C|^2=2^{2nR}$ for most codes while when $R$ is above this threshold, most codes
have a typical sumset whose size is roughly $|\mathcal C|\cdot
2^{nD}=2^{n(R+D)}$ due to the linear structure of the codes. The threshold $D$
depends solely on the alphabet size $q$ and takes value in $[1/2, \log
\sqrt{e})$. More generally, we completely characterize the asymptotic size of
typical sumsets of two nested linear codes $\mathcal C_1, \mathcal C_2$ with
different rates. As an application of the result, we study the communication
problem where the integer sum of two codewords is to be decoded through a
general two-user multiple-access channel.
