You can use a generalized linear mixed model (binomial family), with person as a random effect and sibling as a fixed effect.
R this would be conducted as
data <- data.frame(person=as.factor(c(1,1,2,2)),
data$failures <- data$attempts - data$hits
fit <- lmer(cbind(hits, failures) ~ sibling + (1 | person), family=binomial(), data=data)
The output gives a (two-sided) p-value of $0.00101$ in this example:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -2.7726 0.2062 -13.449 < 2e-16 ***
sibling1 1.2104 0.3682 3.288 0.00101 **
The positive estimate indicates the siblings do better. This differs from the result of combining a bunch of chi-squared tests, one for each person:
chisq <- by(data, data$person, function(x) chisq.test(cbind(x$failures, x$hits)),
stat <- -2 * sum(unlist(lapply(chisq, function(x) log(x$p.value)))) #$
pchisq(stat, df=2, lower.tail=FALSE)
The separate p-values are $6.9$% and $6.8$%, respectively, which when combined as shown yield a p-value of $0.47$% rather than the GLMM p-value of $0.10$%.
One danger of using the chi-squared tests occurs when the effects for different people are in opposite directions: combining their (two-sided) p-values would be erroneous. There is no such problem with the GLMM.
There is more power and flexibility in using the GLMM compared to conducting the chi-squared tests. Moreover, the GLMM will handle multiple siblings per person and multiple experiments per person without any change; it is difficult to see how to adapt chi-squared contingency table analysis to those generalizations.