Peer leadership is challenging. Siuan's frustrated uttering that "[m]ost of my troubles with the Hall stem from you," is an indication to the nuanced challenges that being in a position of leadership over friends can cause. There is an unspoken expectation of loyalty with friends (otherwise, why have friends?). But there are also aims and goals of the organization, and personal relationships can cloud judgment. Or give the appearance of favoritism. Or both (and more!). In this case, it's Moiraine's lack of communications with Siuan that is causing the frustration. As readers, we know the reason behind - they were on the run since Emond's Field - but for Siuan, she has no knowledge of her friend's intentions. At the moment, it appears that she's covering for a rogue Aes Sedai, for which the punishment would be even harsher if people remembered their friendship as novices.
Lots of organizations have hierarchical structures to put the weight of the position behind the person. The White Tower appears more decentralized, giving Sisters more autonomy and freedom to pursue personal goals or adjust how they achieve the Tower's desires. There is an allegiance to ajah, but there also seems to be a direct line from the Amyrlin Seat to each Aes Sedai. This gives the Amyrlin the ability to give direct orders to hand-picked Aes Sedai, but also places the burden of their performance squarely on her shoulders.
It's a balancing act. Being friendly, but not having favorites. Being approachable, but staying professional. Staying focused on the mission, but open to alternative solutions. Being responsible and accountable for outcomes, but trusting the team to drive the bus.