Here's an example with $|B_1|=1, |B_2|=2$:  Let $G=\mathbb Z*\mathbb Z$ (free product). Write non-identity elements of $G$ multiplicatively as reduced words with $X^{\mathbb Z-\{0\}}, Y^{\mathbb Z-\{0\}} $(with symbols $X,Y$).  Let $A_0=X^{\mathbb Z - 2\mathbb Z}, A_1 = \{$reduced words with $X^{2\mathbb N},Y^{2\mathbb N}\}, A=A_0A_1, B_1=\{1\}, B_2=\{X^2,Y^2\}$.  Note we can't have $A \cdot B_1=A \cdot B_2=G$ with $|B_1|=1, |B_2|>1$.