I have no idea what's being done by the organizations that suffered from Heartbleed in terms of proactive reaching out to victims. This is why security breach notification laws are so important. They would require exactly what you are suggesting. They would require notification to those victims, so that they could take the precautions to shut down accounts and protect themselves against fraud.
A very important point was made earlier. The fraud could happen years later. In fact, there is a growing category of identity fraud in the United States right now involving children, where the fraudsters get hold of young children's social security numbers.
If you choose and some parents do choose to get social insurance numbers at the time of birth—you can for your children—so that they can register them for educational savings plans or whatever. Once you do that it becomes susceptible to identity theft. Someone might not realize until they're 18 years old and they go to get their first job or file their first income tax return that they have been a victim of identity fraud for years on the basis of this previously issued information.
This problem of a lag between the theft and the fraud can be very significant.