e.g. is $\widehat{\mathbf{SET}}$ locally small?

No, not in general. Todd gave a useful general answer and here is a concrete counterexample. Let $\mathbf{C}$ be a category whose objects are sets and where the only morphisms are the identities. Consider the functors $F, G : \mathbf{C} \to \mathbf{Set}$ given by $F(X) = 1 = \lbrace 0 \rbrace$ and $G(X) = 2 = \lbrace 0,1\rbrace$. Because $\mathbf{C}$ is discrete, every family of maps $(\eta_X : F(X) \to G(X))_{X \in \mathbf{C}}$ is natural. Therefore, natural transformations $F \to G$ are in bijective correspondence with classes:
Thus the presheaf category $\widehat{\mathbf{C}}$ is not locally small (note that $\mathbf{C}^{op} = \mathbf{C}$). I cannot think at the moment of concrete presheaves $F$ and $G$ on $\mathbf{Set}$ for which the natural transformations $F \to G$ form a proper class. Someone please help. 


There's a theorem due to Freyd and Street that says that if $C$ is locally small and the presheaf category $Set^{C^{op}}$ is locally small, then $C$ is equivalent to a small category. See this paper. 

