A best test then for identical mean RV's looks at the how much of the area of the distribution is centered vs. at the tails. Higher standard deviation distributions are going to have more area in the tails.
You would first form the single normal rv corresponding to the sum of your 10 X's scaled by 1/10, i.e. the mean of your X's.
Then you have a HO accept region from [-c,c] and an H0 reject region of $(\infty, -c) \cup (c, \infty)$
You need to specify c such that the alpha of the test (false alarm, Type I error, etc.) is 0.05.
This is the region where you've chosen H1 when H0 was actually true. In this case this is simply the integral of the H0 distribution for |x|>c.
Given c you can then find the power of the test as the integral over this same region for the H1 distribution.
Given all this do you think the c's will be equal if the the standard deviations change?