I know that if $A$ and $B$ are group of finite rank $n$, there is an amalgamated product $A\ast_{F_2} B$ of rank $2$. It is know that for every $n$ there exist groups $A$ and $B$ of rank $\ge n$ and suitable injective group morphisms $F_2 \to A$ and $F_2 \to B$ from the free group on two generators such that the amalgamated product $A \ast_{F_2} B$ has rank $2$. So amalgamated products do not behave well with respect to (minimal) cardinality of generators.
But how terrible can it be? Can we find, for instance, non finitely generated $A$ and $B$ (or: at least one of the two is not finitely generated) with $A\ast_C B$ finitely generated? What about the same question for finitely presented groups?
Edit: to be more concrete, I hope that the following is true, but I suspect it is not: if $A \ast_C B$ is finitely presented, then both $A$ and $B$ are finitely generated.