Let's say I have two samples of results of two bernoulli experiments.H0:p1=p2H1:p1≠p2And I want to...

Step 1
Suppose the confidence intervals ${\displaystyle I_j}$ have the property that ${\displaystyle p\in I_j}$ with probability ${\displaystyle \ge 1-\alpha }$.
Then under the null hypothesis, ${\displaystyle p\in I_1\cap I_2}$ with probability ${\displaystyle \ge {(1-\alpha )}^2=1-2\alpha +{\alpha }^2}$, so this is a lower bound for the probability that ${\displaystyle I_1}$ and ${\displaystyle I_2}$ intersect.
Thus if they don't intersect, you can reject ${\displaystyle H_0}$ with confidence level at most ${\displaystyle a\alpha -{\alpha }^2}$.
The true confidence level is presumably better than that, but without more analysis we don't know how much better.

It is true that if the intervals don't overlap, then there is a statistically significant difference. However, the converse is not true. Overlapping intervals do not imply that you cannot reject the null hypothesis.
The best approach to take is the one I'm assuming you've already done: Construct a confidence interval for the difference ${\displaystyle p_1-p_2}$ and check to see if it contains the point 0 or not.