Given,

$S_1\equiv {\left(x+1\right)}^2+{\left(y+2\right)}^2=r^2$ and $S_2\equiv x^2+y^2-4x-4y+4=0$ or $S_2\equiv {\left(x-2\right)}^2+{\left(y-2\right)}^2=2^2$

Now, if two circles intersect at two distinct points then $\left|r_1-r_2\right|<C_1{C}_2<r_1+r_2$.

$\Rightarrow \left|r-2\right|<\sqrt{3^2+4^2}<r+2$

$\Rightarrow \left|r-2\right|<5<r+2$

$\Rightarrow \left|r-2\right|<5 \text{and} r+2>5$

$\Rightarrow -5<r-2<5 \text{and} r>3$

$\Rightarrow -3<r<7 \text{and} r>3$

$\Rightarrow 3<r<7$