In an experiment with binary outcomes, a researcher compares two groups ($n_1=50, n_2=50$) to test for differences in proportions $p_1$ and $p_2$. If the pooled proportion is $\hat{p} = 0.3$, what is the estimated standard error for the difference $\hat{p}_1 - \hat{p}_2$ under the null hypothesis $H_0: p_1 = p_2$?