Several methods for generating random Steiner triple systems (STSs) have been proposed in the literature, such as Stinson's hill-climbing algorithm and Cameron's algorithm, but these are not yet completely understood. Those algorithms, as well as some variants, are here assessed for STSs of both small and large orders. For large orders, the number of occurrences of certain configurations in the constructed STSs are compared with the corresponding expected values of random hypergraphs. Modifications of the algorithms are proposed.