will victims of child sexual abuse become abusers?

It is impossible to predict with certainty which convicted sex offender will reoffend after completing his or her sentence. Predicting which childhood circumstance or trauma will lead someone to commit a sex crime in adulthood is just as impossible.

From the abstract of a study published on the JAMA(Journal of the American Medical Assciation) Network:

The widespread belief that sexually abused children are uniquely at risk to become sex offenders was not supported by prospective empirical evidence. These new findings suggest that early intervention programs should target children with histories of physical abuse and neglect. They also indicate that existing policies and practices specifically directed at future risk for sex offending for sexually abused children may warrant reevaluation.

That widespread belief can be dangerous.

Not only do children of sex offenders have to endure knowing that a family member is listed on the registry, if those children are victims of the offender, the registry can expose them to the world as victims of sexual abuse. In some jurisdictions, the victim can be identified on the registry as "granddaughter" or "nephew", often enough information to identify the child. 

Imagine growing up having been sexually abused and living in a world that has a widespread belief that you will abuse a child yourself. 

Children who have been sexually abused deserve better than to live under suspicion and fear that they will be abusers themselves.

Via Families Affirming Community Safety (FACTS).

UPDATE:
This is exactly the kind of cruelty that children of registered citizens must endure.

The right of children to be free of the imaginary threat posed by a registered sex offender does not trump the right of other children--children of sex offenders--to have their parents freely take part in school activities with them.