The circles $x^2+y^2+2x-2y+1=0$ and 

$x^2+y^2-2x-2y+1=0$ touch each other

The centres of the two circles are $C_1(-1,1)$ and $C_2(1,1)$ and both have radii equal to 1.

We have,

$C_1{C}_2=2$ and, Sum of the radii $= 2.$

So, the two circles touch each other externally. The equation of the common tangent is obtained by subtracting the two equations.

The equation of the common tangent is

$4x=0\Rightarrow x=0$

Putting$x = 0$ in the equation of the either circle, we get

$y^2-2y+1=0\Rightarrow (y-1{)}^2=0\Rightarrow y=1$

Hence, the points where the two circles touch is $(0, 1 )$