This is a standard example illustrating the notion that conditional independence does not imply unconditional independence. If the experiment consists of choosing a coin at random and then tossing it $n > 1$ times, then the results of the $n$ tosses are conditionally independent observations of a $\text{Bernoulli}(p)$ random variable where $p$ is the parameter for the coin chosen, and thus the total number of Heads is (conditionally)
a $\text{Binomial}(n,p)$ random variable. But, the $n$ tosses are not unconditionally independent.
More explicitly, with two coins with $P(\text{Heads}) = p_i$ and $n=2$, the conditional probabilities of a Head on each trial is $p_i$ and the two tosses are conditionally
independent. But, the law of total probability says that
$$P(\text{Heads}) = \frac{p_1+p_2}{2}$$ on each toss, whereas for the two trials,
it also says that
$$P(2 ~\text{Heads on}~2~\text{tosses}) = \frac{p_1^2+p_2^2}{2}
\neq \left(\frac{p_1+p_2}{2}\right)^2$$
showing that the two tosses are not unconditionally independent.
In short, if the experiment consists of choosing a fresh coin for
each toss, then the trials are independent and the result is
equivalent to tossing a coin with probability $\bar{p}$ where
$\bar{p}$ is the population mean, but this is not the case
if the experiment consists of choosing a coin and tossing it
several times. If the OP is receiving data about "big batches
of trials, not individual ones" it is important to look carefully
at the conditions of the experiment that resulted in the big
batches of data.