A classical result states that internality and idempotence are equivalent for aggregation functions. In the context of data analysis, it is natural to consider random variables as the inputs of the aggregation. In this direction, aggregations of random variables are functions that, given a random vector, return a random variable fulfilling monotonicity and some boundary conditions with respect to a stochastic order. This paper is focused on the definition of different notions of idempotence and internality for aggregations of random variables. The implications between the introduced concepts are studied in detail. In addition, families of aggregations of random variables that fulfill the defined properties are provided.