I don't know any organization that collects and makes available the statistics you're looking for. What follows is an attempt to provide a summary of the statistics I did come across; sprinkled with some references so you have a point of departure for your own research. I'm not sure if you will find what you're looking for, though, since your question seems to be somewhat at odds with the data most organizations collect.
Also, I'd suggest you approach the safety of your child from another angle (assuming you're mainly worried about sexual abuse): I don't think trying to profile people via statistics will work very well, especially because it's been shown over and over that parents are notoriously bad at detecting abusers (this is actually one of the reasons why children often don't disclose the abuse - they don't think the parents would believe them if they disclosed who the abuser is - and sadly, they are often right). Instead, you could look at how child sexual abusers gain the trust of the child and his/her parents, how they bring the child to agree to the abuse and how they keep the child from disclosing it. I found the first reference I give (Sanderson 2004) very useful for this.
With that said, statistics look different depending on a) the age of the child and b) the type of abuse (sexual/physical/emotional/neglect). Also, an important thing to note is that all studies I came across assume that there is a very large percentage of unreported cases of abuse. For example, (Sanderson 2004, p 81) asserts that only 10% of cases are reported to the authorities (other guesses go up to 30%). The victim's gender may also change the perpetrator's profile, but the data is very unreliable. For example, most studies find that more girls are sexually abused than boys, but they also usually point out that boys are more stigmatized by sexual abuse and are therefore much more reluctant to disclose, so whatever profile you try to construct based on gender is flawed from the start.
For sexual abuse, the biggest risk is the local community, with 87% of sexual assaults committed by someone known to the child (Sanderson 2004, p 84). The abuse often remains undetected for years and involves a number of children. About 60% of the sexual abuse is committed by persons not related to the child, e.g. family friends, babysitters, neighbors etc, according to CAFY. 30% are committed by relatives, 10% by strangers. The ABS Personal Safety Survey (2005) (Australia) found that 13.5% of the abuse was committed by the father or stepfather, 30% by another male relative, 17% by a family friend, 16% by a an acquaintance or neighbour, and 15% by another known person (percentages rounded). Note, though, that for younger children, the abuser is often a family member - kids younger than 6 are abused by family members in 50% of the cases according to Darkness to Light. (Sanderson 2006) also points out the role of family members in sexual abuse - she says that Childline, a phone helpline for kids and teens, said 57% of the sexual abuse reported by children and teens calling the helpline between 2001 and 2002 were perpetrated by family members and another 30% were perpetrated by non-family-members known to the child. Note that 61% of callers where between 12 and 15 years old, 73% were female. So conclusions drawn from this data are likely biased, since young kids are definitely and boys are likely underrepresented in the calls.
BTW, the large majority of perpetrators of sexual abuse are men, but there are a few percent of women abusers; the numbers differ for boy and girl victims. Also, studies won't agree on the exact percentage of women perpetrators, some say 5%, some 10%, some propose 20%.
I've recently come across another study that found that for recent years, the age of the abuser has dropped; at least for the country where the study was done, teenagers and older children now lead as the ones who commit the most sexual abuse of children. I think the study somehow correlated this with the availability of smartphones that could record the abuse. But I can't remember into how many bins the data was sorted and what the study was called. I'll update with a reference when I recall where I read it. To be clear: it doesn't mean that >50% of all sexual abusers are young; rather that of all the bins that were created with age ranges, the largest number of abuse cases were sorted into the young age range. Other sources dispute this finding about young perpetrators, though. (Sanderson 2004, p. 204) corroborates it somewhat by stating "increasingly, reports of children sexually abusing other children are emerging", and Darkness to Light states that as many as 40% of victims are sexually abused by older children/young teens.
Tentative conclusion: There's no data here to determine whether people who are related to other kids are safer for your children than people without such ties, but it's unlikely, since there is a relevant percentage of parents who abuse their own children, so I see no reason to assume being a parent will somehow make you immune to being a sexual abuser, and the same goes for other relatives of children, since there's enough evidence that they do prey on their own young relatives, so why wouldn't they prey on children who are not related to them? Access and opportunity seem to win out over the "deterrent" of having kids as relatives.
But at least the statistics indicate that non-parents (older children and teens, for example) provide a higher share of the abusers than parents.
Physical abuse is mostly done by the primary care givers (e.g. father and mother) of the child. So, you could say that children are most at risk to be abused by their own parents/caregivers, since physical abuse, AFAIK (see for example Prevalence of different forms of child maltreatment, but note the data is for Taiwan...), is much more prevalent than sexual abuse. According to the ABS, 55.6% experienced physical abuse from their father or stepfather and 25.9% from their mother or stepmother; a British study came to a somewhat different conclusion (about 49% mothers vs 40% fathers), but according to the Australian Institute of Family Studies, this is because mothers spend so much more time with kids than fathers. If you normalized this data, you might end up with something close to the Australian data. The abuse fathers deal out is more severe, though.
There are other types of abuse besides physical and sexual abuse; for example neglect and emotional abuse (which seems to be the most common type of abuse by far, with sexual abuse the least common type). They have their own statistics, which again make the primary caregivers look bad, but I assume you were primarily interested in CSA and physical abuse.
Abuse by age
Unfortunately, again, I no longer know where I read the following. When I remember, I'll update with references. Until then, you should take this with a grain of salt - I might not remember correctly.
I think that children are at risk for different types of abuse at different ages. For example, the most dangerous age for physical abuse is with very young children (e.g. toddlers and pre-schoolers). This may depend on the definition of "most dangerous", though; lethal abuse by caregivers most often happens at a very young age (see 5), so this may skew the stats when you're primarily interested in how prevalent abuse is at a given age.
The most dangerous age for sexual abuse is for school-age kids 5 to 12 years old (Sanderson 2006). That doesn't mean that very young kids are safe from sexual abuse, but it changes the profile of the abuser (many family members and very young perps for victims under 6, more non-related persons for school-age victims) (see 6).