This is what's at stake here -- not only the possibility that some women may die because their cancers go undetected, but that many others will lose months or years of their lives to debilitating treatments for radiation-caused cancers or, possibly, cancers that didn't require treatment at all.
Of course, any woman who tragically learns from a mammogram that she has breast cancer may regret getting mammograms if she also learns that they were the cause of her cancer. Nonetheless, it remains true that no woman would be wise to avoid mammograms if they are nearly guaranteed to detect cancer ex-ante. In other words, if all women have some reasonably high probability of getting breast cancer, and mammograms cheaply detect cancer while causing a miniscule proportion of the cancers that they detect, mammograms and guaranteed early detection remain a better choice than death from cancer not caused by mammograms, and not detected by mammograms.
Optimally, women should keep getting mammograms until the cost of getting another mammogram equals the benefit of getting another mammogram. The miniscule cost imposed by radiation from mammograms shouldn't be expected to significantly impact the optimal number of mammograms that women get. In other words, if the cost of each mammograms is m, and the benefit of mammogram number f(n), while the expected cost due to radiation exposure from a mammogram is given by p, then we have:
Before women know that radiation from mammograms can be harmful: f(n) = m at optimal
After women know that radiation can be harmful: f(n) = m + p at optimal n. When p is very small, however, the optimal n changes little from that derived in f(n) = m. This will be the case for reasonable assumptions on the benefits of each additional mammogram to women.