Motivated by online labor markets, we consider the online assortment optimization problem faced by a two-sided matching platform that hosts a set of suppliers waiting to match with a customer. Arriving customers are shown an assortment of suppliers and may choose to issue a match request to one of them. After spending some time on the platform, each supplier reviews all the match requests she has received and, based on her preferences, she chooses whether to match with a customer or to leave unmatched. We study how platforms should design online assortment algorithms to maximize the expected number of matches in such two-sided settings. We establish that a simple greedy algorithm is 1/2-competitive against an optimal clairvoyant algorithm that knows in advance the full sequence of customers’ arrivals. However, unlike related online assortment problems, no randomized algorithm can achieve a better competitive ratio, even in asymptotic regimes. To advance beyond this general impossibility, we consider structured settings where suppliers’ preferences are described by the multinomial logit and nested logit choice models. We develop new forms of balancing algorithms, which we call preference-aware, that leverage structural information about suppliers’ choice models to design the associated discount function. In certain settings, these algorithms attain competitive ratios probably larger than the standard “barrier” of 1 -1=e in the adversarial arrival model. Our results suggest that the shape and timing of suppliers’ choices play critical roles in designing online assortment algorithms for two-sided matching platforms.