We study growth rates of infinite and finite sumset patterns in sets of positive density. In the infinite setting, we show that no such rate exists, answering a question of Kra, Moreira, Ritcher, and Robertson. Namely, for any proposed growth rate $\mathcal{H}: \mathbb{N} \to \mathbb{N}$ tending to infinity, we construct a set $A$ of lower density $1$ such that whenever $B,C \subseteq \mathbb{N}$ are infinite and $B+C \subseteq A$ we have the minimum of $|B\cap [N]|$ and $|C \cap [N]|$ is less than $\mathcal{H}(N)$ for infinitely many $N$. In the finitary setting, we prove that for all $δ\in (0,1)$, for all sufficiently large $N$, for all subsets $A$ of $\{1,\dots,N\}$ of proportion $δ$, one can always find sumset patterns $B+C\subseteq A$ with $|B|$ and $|C|$ of order $\log N$, partially resolving a conjecture of Kra, Moreira, Richter, and Robertson. Moreover, we generalize our second result to the case of the $k$-fold sum $B_1 + B_2 + \ldots + B_k \subseteq A$.